Chapter 3 - Trigonometric Functions

Beginner

Topper

1. Introduction - Angles Direction and Measurement

Angle is formed when a ray rotates from initial side to terminal side. Anticlockwise rotation positive angle, clockwise negative. Like clock: anticlockwise is positive, clockwise negative. Example 90° anticlockwise positive, 90° clockwise -90°.

Two systems to measure angles: Degree and Radian. Degree: one full circle 360°, half circle 180°, quarter 90°. Radian: one full circle 2π radians, half circle π radians, quarter π/2 radians. Relationship: 180° = π radians. That's master key.

Real life: steering wheel rotation, fan rotation, cricket ball spin angles, clock hands angle between hour and minute.

Angle ray rotates initial to terminal Anticlockwise positive Clockwise negative 90° anticlockwise positive 90° clockwise -90° | Degree 360° full circle 180° half 90° quarter Radian 2π full π half π/2 quarter Relationship 180°=π radians master key | Real life steering wheel fan rotation cricket ball spin clock hands angle hour minute

Visual Learning Method: Angle ray rotates initial terminal anticlockwise positive clockwise negative, degree 360° full 180° half 90° quarter radian 2π full π half π/2 quarter 180°=π radians master key, steering wheel fan rotation cricket ball spin clock hands hour minute.

Daily Analogy: Angle like opening a door from closed initial to open terminal anticlockwise opening positive clockwise closing negative. Degree like slices of pizza 360 slices full circle radian like length of crust 2π crust full circle.

Introduction - Angles

Revision Points - Easy:

Angle ray rotates initial to terminal anticlockwise positive clockwise negative.
Degree 360° full circle 180° half 90° quarter Radian 2π full π half π/2 quarter.
Relationship 180°=π radians master key.
Real life steering wheel fan rotation cricket ball spin clock hands.

2. Degree Radian Conversion and Arc Length l=rθ

Conversion formulas: To convert degree to radian multiply by π/180. Example 90° to radian 90×π/180=π/2 rad. 60° to radian 60×π/180=π/3 rad. To convert radian to degree multiply by 180/π. Example π/2 rad to degree π/2×180/π=90°. π/3 rad =60°.

Common values must remember: 0°=0 rad, 30°=π/6 rad, 45°=π/4 rad, 60°=π/3 rad, 90°=π/2 rad, 180°=π rad, 270°=3π/2 rad, 360°=2π rad.

Arc length formula: l = rθ where θ in radians, r radius, l arc length. Example circle radius 7 cm angle π/2 rad 90° arc length l=7×π/2=7×3.14/2≈11 cm. If θ in degrees convert first to radian then apply l=rθ. This formula used in engineering wheels distance covered.

Real life: cycle wheel radius 35 cm rotates angle 2π one full rotation distance covered l=rθ=35×2π=70π≈220 cm one rotation circumference. So l=rθ gives circumference 2πr when θ=2π full circle.

Conversion degree to radian multiply π/180 Example 90°×π/180=π/2 60°×π/180=π/3 radian to degree multiply 180/π Example π/2×180/π=90° π/3=60° | Common 0°=0 30°=π/6 45°=π/4 60°=π/3 90°=π/2 180°=π 270°=3π/2 360°=2π must remember | Arc length l=rθ θ in radians r radius l arc length Example r=7 cm θ=π/2 90° l=7×π/2≈11 cm θ in degrees convert to radian first then l=rθ engineering wheels distance | Real life cycle wheel r=35 cm rotates 2π one full rotation distance l=rθ=35×2π=70π≈220 cm one rotation circumference 2πr when θ=2π full circle

Visual Learning Method: Degree to radian multiply π/180 90°×π/180=π/2 60°×π/180=π/3 radian to degree multiply 180/π π/2×180/π=90° π/3=60°, common 0°=0 30°=π/6 45°=π/4 60°=π/3 90°=π/2 180°=π 270°=3π/2 360°=2π must remember, arc length l=rθ θ radians r radius l arc length r=7 θ=π/2 l=7×π/2≈11 cm convert degrees to radian first, cycle wheel r=35 θ=2π l=35×2π=70π≈220 cm one rotation circumference 2πr θ=2π full circle.

Daily Analogy: Degree to radian conversion like rupees to dollars exchange rate π/180. Arc length l=rθ like pizza crust length = radius × angle crust longer if bigger radius or bigger angle.

Degree Radian Conversion l=rθ

$Conversion$

Revision Points - Easy:

Degree to radian multiply π/180 90°=π/2 60°=π/3 radian to degree multiply 180/π π/2=90° π/3=60°.
Common 0°=0 30°=π/6 45°=π/4 60°=π/3 90°=π/2 180°=π 270°=3π/2 360°=2π must remember.
Arc length l=rθ θ in radians r radius l arc length r=7 θ=π/2 l≈11 cm convert degrees to radian first.
Cycle wheel r=35 θ=2π l=35×2π=70π≈220 cm one rotation circumference 2πr when θ=2π.

3. Trigonometric Functions - Unit Circle Definitions

Unit circle radius 1 centered at origin. Point P on circle makes angle θ with positive x-axis coordinates (x,y). Then definitions: cosθ = x/r = x/1 = x, sinθ = y/r = y/1 = y, tanθ = y/x = sinθ/cosθ. So cos is x-coordinate, sin is y-coordinate on unit circle.

Right triangle: opposite = side opposite angle, adjacent = side next to angle, hypotenuse = longest side opposite 90°. sinθ=opposite/hypotenuse tells how high, cosθ=adjacent/hypotenuse tells how far right/left, tanθ=opposite/adjacent = slope.

Other three: cosecθ=1/sinθ=hypotenuse/opposite, secθ=1/cosθ=hypotenuse/adjacent, cotθ=1/tanθ=adjacent/opposite = cosθ/sinθ.

Identity: sin²θ+cos²θ=1 always because x²+y²=r²=1²=1 unit circle. Divide by cos²: 1+tan²=sec². Divide by sin²: 1+cot²=cosec².

Unit circle radius 1 origin Point P angle θ with positive x-axis coordinates (x,y) cosθ=x/r=x/1=x sinθ=y/r=y/1=y tanθ=y/x=sinθ/cosθ cos x-coordinate sin y-coordinate unit circle | Right triangle opposite side opposite angle adjacent next to angle hypotenuse longest opposite 90° sinθ=opposite/hypotenuse how high cosθ=adjacent/hypotenuse how far right/left tanθ=opposite/adjacent slope | cosecθ=1/sinθ=hypotenuse/opposite secθ=1/cosθ=hypotenuse/adjacent cotθ=1/tanθ=adjacent/opposite=cosθ/sinθ | Identity sin²+cos²=1 always x²+y²=r²=1²=1 unit circle Divide cos² 1+tan²=sec² Divide sin² 1+cot²=cosec²

Visual Learning Method: Unit circle radius 1 origin point P angle θ positive x-axis coordinates (x,y) cosθ=x/r=x sinθ=y/r=y tanθ=y/x=sinθ/cosθ cos x-coordinate sin y-coordinate, right triangle opposite opposite angle adjacent next hypotenuse longest 90° sin opposite/hypotenuse how high cos adjacent/hypotenuse how far right/left tan opposite/adjacent slope, cosec 1/sin hypotenuse/opposite sec 1/cos hypotenuse/adjacent cot 1/tan adjacent/opposite cos/sin, identity sin²+cos²=1 x²+y²=r²=1²=1 unit circle 1+tan²=sec² 1+cot²=cosec².

Daily Analogy: Unit circle like clock face radius 1 cos is horizontal position x sin vertical position y as hand rotates. Right triangle like ladder against wall opposite height sin tells how high ladder reaches cos tells how far base from wall.

Unit Circle Definitions

Revision Points - Easy:

Unit circle radius 1 origin P angle θ coordinates (x,y) cosθ=x sinθ=y tanθ=y/x=sinθ/cosθ.
Right triangle sin opposite/hypotenuse cos adjacent/hypotenuse tan opposite/adjacent slope.
cosec 1/sin hypotenuse/opposite sec 1/cos hypotenuse/adjacent cot 1/tan adjacent/opposite cos/sin.
Identity sin²+cos²=1 x²+y²=r²=1²=1 1+tan²=sec² 1+cot²=cosec².

4. Signs in Quadrants ASTC and Domain Range

ASTC trick All Students Take Calculus All Silver Tea Cups: Quadrant I All positive sin cos tan all positive 0°-90°, Quadrant II Sin positive sin and cosec positive rest negative 90°-180°, Quadrant III Tan positive tan and cot positive rest negative 180°-270°, Quadrant IV Cos positive cos and sec positive rest negative 270°-360°. Trick: All positive I, Sin positive II, Tan positive III, Cos positive IV.

Domain Range: sin Domain R all real angles Range [-1,1] because sin is y-coordinate on unit circle y between -1 and 1. cos Domain R Range [-1,1] x-coordinate between -1 and 1. tan Domain R - (2n+1)π/2 because tan=sin/cos cos=0 at 90° 270° etc undefined Range R all real (-∞,∞) because tan can be any value slope can be any. cot Domain R - nπ sin=0 at 0° 180° etc undefined Range R. sec Domain R - (2n+1)π/2 cos=0 undefined Range (-∞,-1]∪[1,∞) because sec=1/cos cos between -1 and 1 so sec ≤-1 or ≥1. cosec Domain R - nπ sin=0 undefined Range (-∞,-1]∪[1,∞).

Real life: ASTC helps find sign of shadow, building height signs.

ASTC All Students Take Calculus All Silver Tea Cups Quadrant I All positive sin cos tan all positive 0°-90° Quadrant II Sin positive sin cosec positive rest negative 90°-180° Quadrant III Tan positive tan cot positive rest negative 180°-270° Quadrant IV Cos positive cos sec positive rest negative 270°-360° Trick All positive I Sin II Tan III Cos IV | Domain Range sin Domain R all real Range [-1,1] y-coordinate unit circle y -1 to 1 cos Domain R Range [-1,1] x -1 to 1 tan Domain R - (2n+1)π/2 tan=sin/cos cos=0 at 90° 270° undefined Range R (-∞,∞) tan any value slope any cot Domain R - nπ sin=0 at 0° 180° undefined Range R sec Domain R - (2n+1)π/2 cos=0 undefined Range (-∞,-1]∪[1,∞) sec=1/cos cos -1 to 1 sec ≤-1 or ≥1 cosec Domain R - nπ sin=0 undefined Range (-∞,-1]∪[1,∞) | Real life ASTC sign shadow building height signs

Visual Learning Method: ASTC All Students Take Calculus All Silver Tea Cups Quadrant I All positive sin cos tan all positive 0°-90° II Sin positive sin cosec positive rest negative 90°-180° III Tan positive tan cot positive rest negative 180°-270° IV Cos positive cos sec positive rest negative 270°-360° Trick All I Sin II Tan III Cos IV, Domain Range sin Domain R Range [-1,1] y -1 to 1 cos Domain R Range [-1,1] x -1 to 1 tan Domain R-(2n+1)π/2 tan=sin/cos cos=0 at 90° 270° undefined Range R (-∞,∞) any value slope any cot Domain R-nπ sin=0 at 0° 180° undefined Range R sec Domain R-(2n+1)π/2 cos=0 undefined Range (-∞,-1]∪[1,∞) sec=1/cos ≤-1 or ≥1 cosec Domain R-nπ sin=0 undefined Range (-∞,-1]∪[1,∞).

Daily Analogy: ASTC like four houses Quadrant I all rich all positive, II only Sin family rich, III only Tan family rich, IV only Cos family rich. Domain like where function lives allowed angles, Range like values function can give sin always between -1 and 1 cannot go beyond like height of unit circle.

Signs Quadrants ASTC Domain Range

Revision Points - Easy:

ASTC All Students Take Calculus All Silver Tea Cups I All positive all sin cos tan II Sin positive sin cosec III Tan positive tan cot IV Cos positive cos sec.
Trick All positive I Sin II Tan III Cos IV.
sin Domain R Range [-1,1] cos Domain R Range [-1,1] tan Domain R-(2n+1)π/2 Range R (-∞,∞) cot Domain R-nπ Range R sec Domain R-(2n+1)π/2 Range (-∞,-1]∪[1,∞) cosec Domain R-nπ Range (-∞,-1]∪[1,∞).

5. Graphs - Sine Cosine Tangent Waves Periodicity

Graph of sin x wave starts at 0 goes up to +1 at π/2 90° down to 0 at π 180° down to -1 at 3π/2 270° back to 0 at 2π 360° repeats every 2π period 2π amplitude 1 height from center to peak 1. So sin(x+2π)=sin x repeats every 2π.

Graph of cos x wave starts at +1 at 0° goes down to 0 at π/2 down to -1 at π up to 0 at 3π/2 up to +1 at 2π same shape as sin but shifted left by π/2 90° cos x = sin(x+π/2). Period 2π amplitude 1. cos(x+2π)=cos x repeats every 2π.

Graph of tan x has no amplitude grows without limit vertical asymptotes where cos=0 at x=π/2+nπ 90°+n180° undefined tan goes to ±∞ crosses 0 at x=0,π,2π etc Period π not 2π tan(x+π)=tan x repeats every π. No max min because grows without limit.

Periodicity: function repeats after interval period. sin and cos period 2π 360°, tan and cot period π 180°, sec and cosec period 2π.

Graph sin x wave starts 0 up +1 at π/2 90° down 0 at π 180° down -1 at 3π/2 270° back 0 at 2π 360° repeats every 2π period 2π amplitude 1 height center to peak 1 sin(x+2π)=sin x repeats every 2π | Graph cos x wave starts +1 at 0° down 0 at π/2 down -1 at π up 0 at 3π/2 up +1 at 2π same shape as sin shifted left by π/2 90° cos x=sin(x+π/2) Period 2π amplitude 1 cos(x+2π)=cos x repeats every 2π | Graph tan x no amplitude grows without limit vertical asymptotes where cos=0 at x=π/2+nπ 90°+n180° undefined tan ±∞ crosses 0 at x=0,π,2π Period π not 2π tan(x+π)=tan x repeats every π No max min grows without limit | Periodicity repeats after interval period sin cos period 2π 360° tan cot period π 180° sec cosec period 2π

Visual Learning Method: Sin wave starts 0 up +1 at π/2 90° down 0 at π 180° down -1 at 3π/2 270° back 0 at 2π 360° repeats every 2π period 2π amplitude 1 center to peak 1 sin(x+2π)=sin x repeats every 2π, cos wave starts +1 at 0° down 0 at π/2 down -1 at π up 0 at 3π/2 up +1 at 2π same shape as sin shifted left π/2 90° cos x=sin(x+π/2) Period 2π amplitude 1 cos(x+2π)=cos x repeats every 2π, tan no amplitude grows without limit vertical asymptotes cos=0 at π/2+nπ 90°+n180° undefined tan ±∞ crosses 0 at 0,π,2π Period π tan(x+π)=tan x repeats every π No max min grows without limit, periodicity repeats after interval period sin cos 2π 360° tan cot π 180° sec cosec 2π.

Daily Analogy: Sin cos waves like ocean waves repeating up down amplitude height period length until repeat. Sin starts at sea level 0 goes up, cos starts at peak +1 like starting at top of wave. Tan like roller coaster with vertical asymptotes where track goes infinitely up no amplitude no max.

Graphs Sine Cosine Tangent Waves

Revision Points - Easy:

Sin wave starts 0 up +1 at π/2 down 0 at π down -1 at 3π/2 back 0 at 2π repeats every 2π period 2π amplitude 1 sin(x+2π)=sin x.
Cos wave starts +1 at 0 down 0 at π/2 down -1 at π up 0 at 3π/2 up +1 at 2π same shape as sin shifted left π/2 cos x=sin(x+π/2) Period 2π amplitude 1.
Tan no amplitude grows without limit vertical asymptotes at π/2+nπ 90°+n180° undefined ±∞ crosses 0 at 0,π,2π Period π tan(x+π)=tan x No max min.
Periodicity repeats after interval sin cos period 2π tan cot period π sec cosec period 2π.

6. Trigonometric Identities - Sum Difference Allied Angles

Basic identities must remember: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ.

Negative angles: sin(-θ)=-sinθ odd, cos(-θ)=cosθ even, tan(-θ)=-tanθ odd. So sin is odd cos even.

Allied angles 90°±θ 180°±θ 270°±θ 360°±θ: Trick: If angle is 90° or 270° (π/2,3π/2) function changes sin↔cos tan↔cot sec↔cosec, if angle is 180° or 360° (π,2π) function stays same sin stays sin cos stays cos. Then sign decided by quadrant where original angle lies using ASTC.

Examples: sin(90°+θ)=cosθ? Quadrant II 90°+θ in II Sin positive so sin(90°+θ)=+cosθ. cos(90°+θ)= -sinθ because Quadrant II Cos negative and function changes cos→sin so -sinθ. sin(180°+θ)= -sinθ Quadrant III Sin negative function stays sin so -sinθ. cos(180°+θ)= -cosθ Quadrant III Cos negative stays cos -cosθ. sin(270°+θ)= -cosθ Quadrant IV Sin negative changes sin→cos -cosθ. tan(90°+θ)= -cotθ Quadrant II Tan negative changes tan→cot -cotθ.

Sum difference formulas: sin(A+B)=sinA cosB + cosA sinB, sin(A-B)=sinA cosB - cosA sinB, cos(A+B)=cosA cosB - sinA sinB, cos(A-B)=cosA cosB + sinA sinB, tan(A+B)=(tanA+tanB)/(1-tanA tanB), tan(A-B)=(tanA-tanB)/(1+tanA tanB).

Double angle: sin2A=2 sinA cosA, cos2A=cos²A-sin²A=2cos²A-1=1-2sin²A, tan2A=2tanA/(1-tan²A).

Basic sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² must remember | Negative angles sin(-θ)=-sinθ odd cos(-θ)=cosθ even tan(-θ)=-tanθ odd sin odd cos even | Allied angles 90°±θ 180°±θ 270°±θ 360°±θ Trick If angle 90° or 270° π/2 3π/2 function changes sin↔cos tan↔cot sec↔cosec If angle 180° or 360° π 2π function stays same sin stays sin cos stays cos Then sign decided by quadrant original angle lies ASTC | Examples sin(90°+θ)=cosθ Quadrant II 90°+θ II Sin positive sin(90°+θ)=+cosθ cos(90°+θ)=-sinθ Quadrant II Cos negative changes cos→sin -sinθ sin(180°+θ)=-sinθ Quadrant III Sin negative stays sin -sinθ cos(180°+θ)=-cosθ Quadrant III Cos negative stays cos -cosθ sin(270°+θ)=-cosθ Quadrant IV Sin negative changes sin→cos -cosθ tan(90°+θ)=-cotθ Quadrant II Tan negative changes tan→cot -cotθ | Sum difference sin(A+B)=sinA cosB+cosA sinB sin(A-B)=sinA cosB-cosA sinB cos(A+B)=cosA cosB-sinA sinB cos(A-B)=cosA cosB+sinA sinB tan(A+B)=(tanA+tanB)/(1-tanA tanB) tan(A-B)=(tanA-tanB)/(1+tanA tanB) | Double angle sin2A=2 sinA cosA cos2A=cos²A-sin²A=2cos²A-1=1-2sin²A tan2A=2tanA/(1-tan²A)

Visual Learning Method: Basic sin²+cos²=1 1+tan²=sec² 1+cot²=cosec², negative sin(-θ)=-sinθ odd cos(-θ)=cosθ even tan(-θ)=-tanθ odd sin odd cos even, allied angles 90°±θ 180°±θ 270°±θ 360°±θ trick If 90° or 270° π/2 3π/2 function changes sin↔cos tan↔cot sec↔cosec If 180° or 360° π 2π function stays same sin stays sin cos stays cos sign quadrant ASTC Examples sin(90°+θ)=cosθ Quadrant II Sin positive +cosθ cos(90°+θ)=-sinθ Quadrant II Cos negative changes -sinθ sin(180°+θ)=-sinθ Quadrant III Sin negative stays -sinθ cos(180°+θ)=-cosθ Quadrant III Cos negative stays -cosθ sin(270°+θ)=-cosθ Quadrant IV Sin negative changes -cosθ tan(90°+θ)=-cotθ Quadrant II Tan negative changes -cotθ, sum difference sin(A+B)=sinA cosB+cosA sinB sin(A-B)=sinA cosB-cosA sinB cos(A+B)=cosA cosB-sinA sinB cos(A-B)=cosA cosB+sinA sinB tan(A+B)=(tanA+tanB)/(1-tanA tanB) tan(A-B)=(tanA-tanB)/(1+tanA tanB), double sin2A=2 sinA cosA cos2A=cos²-sin²=2cos²-1=1-2sin² tan2A=2tanA/(1-tan²A).

Daily Analogy: Allied angles trick like 90° and 270° are transformers change function sin becomes cos, 180° and 360° are non-transformers stay same sin stays sin sign decided by quadrant house where angle lives using ASTC All Sin Tan Cos houses.

Trigonometric Identities

Revision Points - Easy:

Basic sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² must remember.
Negative sin(-θ)=-sinθ odd cos(-θ)=cosθ even tan(-θ)=-tanθ odd.
Allied angles 90°±θ 270°±θ function changes sin↔cos tan↔cot sec↔cosec 180°±θ 360°±θ function stays same sin stays sin cos stays cos sign decided by quadrant ASTC.
Examples sin(90°+θ)=cosθ cos(90°+θ)=-sinθ sin(180°+θ)=-sinθ cos(180°+θ)=-cosθ sin(270°+θ)=-cosθ tan(90°+θ)=-cotθ.
Sum difference sin(A+B)=sinA cosB+cosA sinB sin(A-B)=sinA cosB-cosA sinB cos(A+B)=cosA cosB-sinA sinB cos(A-B)=cosA cosB+sinA sinB tan(A+B)=(tanA+tanB)/(1-tanA tanB) tan(A-B)=(tanA-tanB)/(1+tanA tanB).
Double sin2A=2 sinA cosA cos2A=cos²-sin²=2cos²-1=1-2sin² tan2A=2tanA/(1-tan²A).

7. Real Life Applications - Height Distance Waves

Height and distance: Use tanθ = height/distance to find height of building tree tower using angle of elevation. Example person 20 m away from building angle of elevation 45° height = distance×tan45°=20×1=20 m building height 20 m. Shadow: tanθ = height/shadow length.

Waves: Sound waves light waves radio waves water waves all described by sin and cos functions y=A sin(ωt) amplitude A frequency ω. Music notes are sin waves different frequencies.

Engineering: AC current voltage V=V₀ sin(ωt) alternating current, bridges building stress analysis uses trig, navigation GPS uses trigonometry to find distance between places lat long, astronomy distance of stars planets uses parallax trig.

Construction: Roof slope angle tanθ = rise/run, ramp angle for wheelchair.

Real life: Ferris wheel height as function of angle sin, tides high low tide sin wave, pendulum swing sin wave, circular motion projection sin cos.

Height distance tanθ=height/distance height building tree tower angle of elevation Example 20 m away building angle 45° height=distance×tan45°=20×1=20 m building height 20 m Shadow tanθ=height/shadow length | Waves Sound light radio water all described sin cos y=A sin(ωt) amplitude A frequency ω Music notes sin waves different frequencies | Engineering AC current voltage V=V₀ sin(ωt) alternating current bridges building stress analysis trig navigation GPS distance places lat long astronomy distance stars planets parallax trig | Construction Roof slope tanθ=rise/run ramp angle wheelchair | Real life Ferris wheel height as function angle sin tides high low tide sin wave pendulum swing sin wave circular motion projection sin cos

Visual Learning Method: Height distance tanθ=height/distance building tree tower angle of elevation 20 m away angle 45° height=distance×tan45°=20×1=20 m building height 20 m shadow tanθ=height/shadow length, waves sound light radio water all described sin cos y=A sin(ωt) amplitude A frequency ω music notes sin waves different frequencies, engineering AC current voltage V=V₀ sin(ωt) alternating current bridges building stress analysis trig navigation GPS distance lat long astronomy distance stars planets parallax trig, construction roof slope tanθ=rise/run ramp angle wheelchair, Ferris wheel height as function angle sin tides high low tide sin wave pendulum swing sin wave circular motion projection sin cos.

Daily Analogy: Height and distance like measuring building height using shadow and angle sun tanθ = height/shadow like using triangle to measure tall thing without climbing. Waves like music sin waves different pitches different frequencies high frequency high pitch. Ferris wheel height goes up down sin wave as wheel rotates.

Real Life Applications

Revision Points - Easy:

Height distance tanθ=height/distance height building tree tower angle of elevation 20 m away angle 45° height=20×tan45°=20×1=20 m shadow tanθ=height/shadow length.
Waves sound light radio water all described sin cos y=A sin(ωt) amplitude A frequency ω music notes sin waves different frequencies.
Engineering AC current voltage V=V₀ sin(ωt) alternating current bridges building stress analysis navigation GPS distance lat long astronomy distance stars planets parallax trig.
Construction roof slope tanθ=rise/run ramp angle wheelchair Ferris wheel height as function angle sin tides high low tide sin wave pendulum swing sin wave circular motion projection sin cos.

8. Formulas Short Tricks Gold Table - Exam Ready

Must remember table values 0° 30° 45° 60° 90°:

sin 0°=0 sin30°=1/2 sin45°=√2/2≈0.707 sin60°=√3/2≈0.866 sin90°=1. Trick: √0/2=0 √1/2=1/2 √2/2 √3/2 √4/2=1 sequence √n/2 n=0,1,2,3,4.

cos 0°=1 cos30°=√3/2 cos45°=√2/2 cos60°=1/2 cos90°=0 reverse of sin √4/2 to √0/2.

tan 0°=0 tan30°=1/√3≈0.577 tan45°=1 tan60°=√3≈1.732 tan90°=∞ undefined.

Hand trick: hold left hand palm facing you fingers represent angles 0° little finger 30° ring 45° middle 60° index 90° thumb count fingers below angle number of fingers below divide sqrt by 2 gives sin. Example 30° one finger below √1/2=1/2 sin30°=1/2.

Formulas list: sin²+cos²=1, 1+tan²=sec², 1+cot²=cosec², sin(A±B), cos(A±B), tan(A±B), sin2A, cos2A, tan2A, sin3A=3sinA-4sin³A cos3A=4cos³A-3cosA.

Short trick: sin is odd cos even, 90° 270° change function 180° 360° stay same sign by ASTC quadrant, period sin cos 2π tan cot π.

Table must remember 0° 30° 45° 60° 90° sin 0°=0 sin30°=1/2 sin45°=√2/2≈0.707 sin60°=√3/2≈0.866 sin90°=1 Trick √0/2=0 √1/2=1/2 √2/2 √3/2 √4/2=1 sequence √n/2 n=0,1,2,3,4 | cos 0°=1 cos30°=√3/2 cos45°=√2/2 cos60°=1/2 cos90°=0 reverse sin √4/2 to √0/2 | tan 0°=0 tan30°=1/√3≈0.577 tan45°=1 tan60°=√3≈1.732 tan90°=∞ undefined | Hand trick hold left hand palm facing you fingers angles 0° little 30° ring 45° middle 60° index 90° thumb count fingers below angle number fingers below divide sqrt by 2 gives sin Example 30° one finger below √1/2=1/2 sin30°=1/2 | Formulas sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² sin(A±B) cos(A±B) tan(A±B) sin2A cos2A tan2A sin3A=3sinA-4sin³A cos3A=4cos³A-3cosA | Short trick sin odd cos even 90° 270° change function 180° 360° stay same sign by ASTC quadrant period sin cos 2π tan cot π

Visual Learning Method: Table 0° 30° 45° 60° 90° sin 0°=0 sin30°=1/2 sin45°=√2/2≈0.707 sin60°=√3/2≈0.866 sin90°=1 Trick √0/2=0 √1/2=1/2 √2/2 √3/2 √4/2=1 sequence √n/2 n=0,1,2,3,4, cos 0°=1 cos30°=√3/2 cos45°=√2/2 cos60°=1/2 cos90°=0 reverse sin √4/2 to √0/2, tan 0°=0 tan30°=1/√3≈0.577 tan45°=1 tan60°=√3≈1.732 tan90°=∞ undefined, hand trick left hand palm facing you fingers angles 0° little 30° ring 45° middle 60° index 90° thumb count fingers below angle number fingers below divide sqrt by 2 gives sin Example 30° one finger below √1/2=1/2 sin30°=1/2, formulas sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² sin(A±B) cos(A±B) tan(A±B) sin2A cos2A tan2A sin3A=3sinA-4sin³A cos3A=4cos³A-3cosA, short trick sin odd cos even 90° 270° change function 180° 360° stay same sign by ASTC quadrant period sin cos 2π tan cot π.

Daily Analogy: Table values like memorizing phone numbers but trick √n/2 sequence 0,1,2,3,4 easy sin increases from 0 to 1 cos decreases from 1 to 0 opposite. Hand trick like counting fingers to remember sin values left hand fingers represent angles below count sqrt/2 gives sin.

Formulas Short Tricks Table

Revision Points - Easy:

Table 0° 30° 45° 60° 90° sin 0°=0 sin30°=1/2 sin45°=√2/2≈0.707 sin60°=√3/2≈0.866 sin90°=1 Trick √n/2 n=0,1,2,3,4 √0/2=0 √1/2=1/2 √2/2 √3/2 √4/2=1.
cos 0°=1 cos30°=√3/2 cos45°=√2/2 cos60°=1/2 cos90°=0 reverse sin.
tan 0°=0 tan30°=1/√3≈0.577 tan45°=1 tan60°=√3≈1.732 tan90°=∞ undefined.
Hand trick left hand palm facing you fingers 0° little 30° ring 45° middle 60° index 90° thumb count below √/2 gives sin 30° one finger below √1/2=1/2.
Formulas sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² sin(A±B) cos(A±B) tan(A±B) sin2A=2 sinA cosA cos2A=cos²-sin²=2cos²-1=1-2sin² tan2A=2tanA/(1-tan²A) sin3A=3sinA-4sin³A cos3A=4cos³A-3cosA.
Short trick sin odd cos even 90° 270° change function 180° 360° stay same sign by ASTC quadrant period sin cos 2π tan cot π.

Gold Reference Table - Every Concept Formula Trick Real Life

Concept	Formula / Trick / Fact	Real Life Visual / Use
Angle	Ray rotates initial to terminal Anticlockwise positive Clockwise negative 90° anticlockwise positive 90° clockwise -90°	Door opening anticlockwise positive clockwise closing negative steering wheel fan rotation cricket ball spin clock hands
Degree Radian	Degree 360° full circle 180° half 90° quarter Radian 2π full π half π/2 quarter Relationship 180°=π radians master key Conversion degree to radian multiply π/180 Example 90°×π/180=π/2 60°×π/180=π/3 radian to degree multiply 180/π π/2×180/π=90° π/3=60° Common 0°=0 30°=π/6 45°=π/4 60°=π/3 90°=π/2 180°=π 270°=3π/2 360°=2π must remember	Pizza slices degree 360 slices crust length radian 2π crust full circle rupees to dollars exchange rate π/180
Arc Length l=rθ	l=rθ θ in radians r radius l arc length Example r=7 cm θ=π/2 90° l=7×π/2≈11 cm θ in degrees convert to radian first then l=rθ Cycle wheel r=35 cm rotates 2π one full rotation distance l=rθ=35×2π=70π≈220 cm one rotation circumference 2πr when θ=2π full circle	Pizza crust length = radius × angle crust longer if bigger radius or bigger angle cycle wheel distance one rotation circumference 2πr
Unit Circle	Radius 1 origin Point P angle θ with positive x-axis coordinates (x,y) cosθ=x/r=x/1=x sinθ=y/r=y/1=y tanθ=y/x=sinθ/cosθ cos x-coordinate sin y-coordinate unit circle Right triangle opposite opposite angle adjacent next hypotenuse longest opposite 90° sinθ=opposite/hypotenuse how high cosθ=adjacent/hypotenuse how far right/left tanθ=opposite/adjacent slope cosecθ=1/sinθ=hypotenuse/opposite secθ=1/cosθ=hypotenuse/adjacent cotθ=1/tanθ=adjacent/opposite=cosθ/sinθ	Clock face radius 1 cos horizontal position x sin vertical position y hand rotates ladder against wall opposite height sin how high ladder reaches cos how far base from wall
Identity Basic	sin²θ+cos²θ=1 always x²+y²=r²=1²=1 unit circle Divide cos² 1+tan²=sec² Divide sin² 1+cot²=cosec²	x²+y²=r² unit circle
ASTC Signs	All Students Take Calculus All Silver Tea Cups Quadrant I All positive sin cos tan all positive 0°-90° Quadrant II Sin positive sin cosec positive rest negative 90°-180° Quadrant III Tan positive tan cot positive rest negative 180°-270° Quadrant IV Cos positive cos sec positive rest negative 270°-360° Trick All positive I Sin II Tan III Cos IV	Four houses I all rich all positive II only Sin family rich III only Tan family rich IV only Cos family rich
Domain Range	sin Domain R Range [-1,1] y-coordinate -1 to 1 cos Domain R Range [-1,1] x -1 to 1 tan Domain R-(2n+1)π/2 cos=0 at 90° 270° undefined Range R (-∞,∞) any value slope any cot Domain R-nπ sin=0 at 0° 180° undefined Range R sec Domain R-(2n+1)π/2 cos=0 undefined Range (-∞,-1]∪[1,∞) sec=1/cos ≤-1 or ≥1 cosec Domain R-nπ sin=0 undefined Range (-∞,-1]∪[1,∞)	Domain where function lives allowed angles Range values function can give sin always between -1 and 1 cannot go beyond height of unit circle
Graph Sin Cos Tan	Sin wave starts 0 up +1 at π/2 down 0 at π down -1 at 3π/2 back 0 at 2π repeats every 2π period 2π amplitude 1 center to peak 1 sin(x+2π)=sin x Cos wave starts +1 at 0 down 0 at π/2 down -1 at π up 0 at 3π/2 up +1 at 2π same shape as sin shifted left π/2 cos x=sin(x+π/2) Period 2π amplitude 1 cos(x+2π)=cos x Tan no amplitude grows without limit vertical asymptotes at π/2+nπ 90°+n180° undefined ±∞ crosses 0 at 0,π,2π Period π tan(x+π)=tan x No max min Periodicity repeats after interval sin cos period 2π 360° tan cot period π 180° sec cosec period 2π	Ocean waves repeating up down amplitude height period length until repeat sin starts sea level 0 goes up cos starts peak +1 top of wave tan roller coaster vertical asymptotes infinitely up no amplitude no max Ferris wheel
Negative Angles	sin(-θ)=-sinθ odd cos(-θ)=cosθ even tan(-θ)=-tanθ odd sin odd cos even	Odd even functions
Allied Angles Trick	90°±θ 180°±θ 270°±θ 360°±θ Trick If angle 90° or 270° π/2 3π/2 function changes sin↔cos tan↔cot sec↔cosec If angle 180° or 360° π 2π function stays same sin stays sin cos stays cos Then sign decided by quadrant original angle lies ASTC Examples sin(90°+θ)=cosθ Quadrant II Sin positive +cosθ cos(90°+θ)=-sinθ Quadrant II Cos negative changes -sinθ sin(180°+θ)=-sinθ Quadrant III Sin negative stays -sinθ cos(180°+θ)=-cosθ Quadrant III Cos negative stays -cosθ sin(270°+θ)=-cosθ Quadrant IV Sin negative changes -cosθ tan(90°+θ)=-cotθ Quadrant II Tan negative changes -cotθ	90° 270° transformers change function sin becomes cos 180° 360° non-transformers stay same sin stays sin sign decided by quadrant house where angle lives ASTC All Sin Tan Cos houses
Sum Difference	sin(A+B)=sinA cosB+cosA sinB sin(A-B)=sinA cosB-cosA sinB cos(A+B)=cosA cosB-sinA sinB cos(A-B)=cosA cosB+sinA sinB tan(A+B)=(tanA+tanB)/(1-tanA tanB) tan(A-B)=(tanA-tanB)/(1+tanA tanB)	Sum difference formulas
Double Triple	sin2A=2 sinA cosA cos2A=cos²A-sin²A=2cos²A-1=1-2sin²A tan2A=2tanA/(1-tan²A) sin3A=3sinA-4sin³A cos3A=4cos³A-3cosA	Double triple angle formulas
Height Distance	tanθ=height/distance height building tree tower angle of elevation Example 20 m away building angle 45° height=distance×tan45°=20×1=20 m building height 20 m Shadow tanθ=height/shadow length	Measuring building height using shadow and angle sun tanθ=height/shadow triangle measure tall thing without climbing person 20 m away angle 45° height 20 m
Waves Engineering	Sound light radio water all described sin cos y=A sin(ωt) amplitude A frequency ω Music notes sin waves different frequencies AC current voltage V=V₀ sin(ωt) alternating current bridges building stress analysis trig navigation GPS distance places lat long astronomy distance stars planets parallax trig Construction Roof slope tanθ=rise/run ramp angle wheelchair Ferris wheel height as function angle sin tides high low tide sin wave pendulum swing sin wave circular motion projection sin cos	Music sin waves different pitches high frequency high pitch AC current alternating V=V₀ sin(ωt) roof slope rise/run ramp wheelchair Ferris wheel height up down sin wave tides sin wave pendulum swing sin wave circular motion projection sin cos
Table Values 0-90	sin 0°=0 sin30°=1/2 sin45°=√2/2≈0.707 sin60°=√3/2≈0.866 sin90°=1 Trick √0/2=0 √1/2=1/2 √2/2 √3/2 √4/2=1 sequence √n/2 n=0,1,2,3,4 cos 0°=1 cos30°=√3/2 cos45°=√2/2 cos60°=1/2 cos90°=0 reverse sin √4/2 to √0/2 tan 0°=0 tan30°=1/√3≈0.577 tan45°=1 tan60°=√3≈1.732 tan90°=∞ undefined Hand trick left hand palm facing you fingers angles 0° little 30° ring 45° middle 60° index 90° thumb count fingers below angle number fingers below divide sqrt by 2 gives sin Example 30° one finger below √1/2=1/2 sin30°=1/2	Memorizing phone numbers trick √n/2 sequence 0,1,2,3,4 easy sin increases 0 to 1 cos decreases 1 to 0 opposite hand trick counting fingers left hand fingers represent angles below count sqrt/2 gives sin 30° one finger below √1/2=1/2
Short Tricks General	sin is odd cos even 90° 270° change function 180° 360° stay same sign by ASTC quadrant period sin cos 2π tan cot π sin is odd cos even function	Odd even 90° 270° change 180° 360° stay same sign ASTC period sin cos 2π tan cot π

Final Exam Ready Revision - Every Concept Formula Trick Real Life:

Angle ray rotates initial to terminal anticlockwise positive clockwise negative 90° anticlockwise positive 90° clockwise -90° degree 360° full circle 180° half 90° quarter radian 2π full π half π/2 quarter relationship 180°=π radians master key conversion degree to radian multiply π/180 90°×π/180=π/2 60°×π/180=π/3 radian to degree multiply 180/π π/2×180/π=90° π/3=60° common 0°=0 30°=π/6 45°=π/4 60°=π/3 90°=π/2 180°=π 270°=3π/2 360°=2π must remember arc length l=rθ θ in radians r radius l arc length r=7 cm θ=π/2 90° l=7×π/2≈11 cm θ in degrees convert to radian first then l=rθ cycle wheel r=35 cm rotates 2π one full rotation distance l=rθ=35×2π=70π≈220 cm one rotation circumference 2πr when θ=2π full circle door opening anticlockwise positive clockwise closing negative pizza slices degree 360 slices crust length radian 2π crust full circle rupees to dollars exchange rate π/180 pizza crust length = radius × angle crust longer if bigger radius or bigger angle cycle wheel distance one rotation circumference 2πr.
Unit circle radius 1 origin point P angle θ with positive x-axis coordinates (x,y) cosθ=x/r=x/1=x sinθ=y/r=y/1=y tanθ=y/x=sinθ/cosθ cos x-coordinate sin y-coordinate unit circle right triangle opposite opposite angle adjacent next hypotenuse longest opposite 90° sinθ=opposite/hypotenuse how high cosθ=adjacent/hypotenuse how far right/left tanθ=opposite/adjacent slope cosecθ=1/sinθ=hypotenuse/opposite secθ=1/cosθ=hypotenuse/adjacent cotθ=1/tanθ=adjacent/opposite=cosθ/sinθ identity sin²+cos²=1 always x²+y²=r²=1²=1 unit circle divide cos² 1+tan²=sec² divide sin² 1+cot²=cosec² clock face radius 1 cos horizontal position x sin vertical position y hand rotates ladder against wall opposite height sin how high ladder reaches cos how far base from wall x²+y²=r² unit circle.
ASTC All Students Take Calculus All Silver Tea Cups Quadrant I All positive sin cos tan all positive 0°-90° Quadrant II Sin positive sin cosec positive rest negative 90°-180° Quadrant III Tan positive tan cot positive rest negative 180°-270° Quadrant IV Cos positive cos sec positive rest negative 270°-360° trick All positive I Sin II Tan III Cos IV domain range sin Domain R all real Range [-1,1] y-coordinate unit circle y -1 to 1 cos Domain R Range [-1,1] x -1 to 1 tan Domain R-(2n+1)π/2 tan=sin/cos cos=0 at 90° 270° undefined Range R (-∞,∞) tan any value slope any cot Domain R-nπ sin=0 at 0° 180° undefined Range R sec Domain R-(2n+1)π/2 cos=0 undefined Range (-∞,-1]∪[1,∞) sec=1/cos cos -1 to 1 sec ≤-1 or ≥1 cosec Domain R-nπ sin=0 undefined Range (-∞,-1]∪[1,∞) four houses I all rich all positive II only Sin family rich III only Tan family rich IV only Cos family rich Domain where function lives allowed angles Range values function can give sin always between -1 and 1 cannot go beyond height of unit circle.
Graph sin x wave starts 0 up +1 at π/2 90° down 0 at π 180° down -1 at 3π/2 270° back 0 at 2π 360° repeats every 2π period 2π amplitude 1 height center to peak 1 sin(x+2π)=sin x repeats every 2π cos x wave starts +1 at 0° down 0 at π/2 down -1 at π up 0 at 3π/2 up +1 at 2π same shape as sin shifted left by π/2 90° cos x=sin(x+π/2) Period 2π amplitude 1 cos(x+2π)=cos x repeats every 2π tan x no amplitude grows without limit vertical asymptotes where cos=0 at x=π/2+nπ 90°+n180° undefined tan goes to ±∞ crosses 0 at x=0,π,2π etc Period π not 2π tan(x+π)=tan x repeats every π No max min because grows without limit periodicity repeats after interval period sin and cos period 2π 360° tan and cot period π 180° sec and cosec period 2π ocean waves repeating up down amplitude height period length until repeat sin starts sea level 0 goes up cos starts peak +1 top of wave tan roller coaster vertical asymptotes infinitely up no amplitude no max Ferris wheel.
Basic identities sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² must remember negative angles sin(-θ)=-sinθ odd cos(-θ)=cosθ even tan(-θ)=-tanθ odd sin is odd cos even allied angles 90°±θ 180°±θ 270°±θ 360°±θ trick If angle is 90° or 270° π/2 3π/2 function changes sin↔cos tan↔cot sec↔cosec if angle is 180° or 360° π 2π function stays same sin stays sin cos stays cos then sign decided by quadrant where original angle lies using ASTC examples sin(90°+θ)=cosθ Quadrant II 90°+θ in II Sin positive so sin(90°+θ)=+cosθ cos(90°+θ)=-sinθ because Quadrant II Cos negative and function changes cos→sin so -sinθ sin(180°+θ)=-sinθ Quadrant III Sin negative function stays sin so -sinθ cos(180°+θ)=-cosθ Quadrant III Cos negative stays cos -cosθ sin(270°+θ)=-cosθ Quadrant IV Sin negative changes sin→cos -cosθ tan(90°+θ)=-cotθ Quadrant II Tan negative changes tan→cot -cotθ sum difference sin(A+B)=sinA cosB+cosA sinB sin(A-B)=sinA cosB-cosA sinB cos(A+B)=cosA cosB-sinA sinB cos(A-B)=cosA cosB+sinA sinB tan(A+B)=(tanA+tanB)/(1-tanA tanB) tan(A-B)=(tanA-tanB)/(1+tanA tanB) double angle sin2A=2 sinA cosA cos2A=cos²A-sin²A=2cos²A-1=1-2sin²A tan2A=2tanA/(1-tan²A) 90° 270° transformers change function sin becomes cos 180° 360° non-transformers stay same sin stays sin sign decided by quadrant house where angle lives using ASTC All Sin Tan Cos houses sum difference formulas double angle formulas.
Real life height distance tanθ=height/distance height building tree tower angle of elevation example 20 m away building angle 45° height=distance×tan45°=20×1=20 m building height 20 m shadow tanθ=height/shadow length waves sound light radio water all described sin cos y=A sin(ωt) amplitude A frequency ω music notes sin waves different frequencies engineering AC current voltage V=V₀ sin(ωt) alternating current bridges building stress analysis trig navigation GPS distance places lat long astronomy distance stars planets parallax trig construction roof slope tanθ=rise/run ramp angle wheelchair Ferris wheel height as function angle sin tides high low tide sin wave pendulum swing sin wave circular motion projection sin cos measuring building height using shadow and angle sun tanθ=height/shadow triangle measure tall thing without climbing person 20 m away angle 45° height 20 m waves music sin waves different pitches high frequency high pitch AC current alternating V=V₀ sin(ωt) roof slope rise/run ramp wheelchair Ferris wheel height up down sin wave tides sin wave pendulum swing sin wave circular motion projection sin cos.
Table must remember 0° 30° 45° 60° 90° sin 0°=0 sin30°=1/2 sin45°=√2/2≈0.707 sin60°=√3/2≈0.866 sin90°=1 trick √0/2=0 √1/2=1/2 √2/2 √3/2 √4/2=1 sequence √n/2 n=0,1,2,3,4 cos 0°=1 cos30°=√3/2 cos45°=√2/2 cos60°=1/2 cos90°=0 reverse of sin √4/2 to √0/2 tan 0°=0 tan30°=1/√3≈0.577 tan45°=1 tan60°=√3≈1.732 tan90°=∞ undefined hand trick hold left hand palm facing you fingers represent angles 0° little finger 30° ring 45° middle 60° index 90° thumb count fingers below angle number of fingers below divide sqrt by 2 gives sin example 30° one finger below √1/2=1/2 sin30°=1/2 formulas sin²+cos²=1 1+tan²=sec² 1+cot²=cosec² sin(A±B) cos(A±B) tan(A±B) sin2A cos2A tan2A sin3A=3sinA-4sin³A cos3A=4cos³A-3cosA short trick sin is odd cos even 90° 270° change function 180° 360° stay same sign by ASTC quadrant period sin cos 2π tan cot π memorizing phone numbers trick √n/2 sequence 0,1,2,3,4 easy sin increases from 0 to 1 cos decreases from 1 to 0 opposite hand trick counting fingers left hand fingers represent angles below count sqrt/2 gives sin 30° one finger below √1/2=1/2.