Inverse Trigonometric Functions
1. What is Inverse Trig? - Undoing Trig
Sine takes angle and gives value: sin 30° = 0.5. Inverse sine does opposite: takes value and gives angle: sin⁻¹(0.5)=30°. But sin is not one-one on full real line, many angles give same value. So we restrict domain to make it one-one and invertible. For sin we take [-π/2,π/2], for cos [0,π], for tan (-π/2,π/2). Then inverse exists. Notation sin⁻¹ x = arcsin x.
Inverse trig undo trig. sin θ = x ⇒ θ=sin⁻¹ x. Need principal branch to make one-one. sin on [-π/2,π/2], cos on [0,π], tan on (-π/2,π/2). Then inverse defined. sin⁻¹: arcsin.
• Trig: angle → value, Inverse: value → angle
• Example: sin 30°=0.5, sin⁻¹(0.5)=30°
• Need restriction because sin not one-one on R
• sin branch [-π/2,π/2], cos [0,π], tan (-π/2,π/2)
• Notation: sin⁻¹=arcsin, cos⁻¹=arccos, tan⁻¹=arctan
2. Graphs and Principal Value Branches
Graph of sin⁻¹ is mirror of sin over y=x, but only for branch [-π/2,π/2]. It starts at (-1,-π/2) goes to (1,π/2) increasing. cos⁻¹ starts at (-1,π) to (1,0) decreasing. tan⁻¹ goes from -∞ to ∞ with horizontal asymptotes -π/2 and π/2, S-shaped. Principal value is angle in principal range. Example principal value of sin⁻¹(√3/2)=π/3 because π/3∈[-π/2,π/2]. sin⁻¹(-1/2)=-π/6.
Graphs: sin⁻¹ increasing [-1,1]→[-π/2,π/2], cos⁻¹ decreasing [-1,1]→[0,π], tan⁻¹ increasing R→(-π/2,π/2) with asymptotes. Principal value branch is the range we fix to make function. Principal value of sin⁻¹ x is angle in [-π/2,π/2] whose sine is x.
• sin⁻¹ graph mirror of sin branch, [-1,1]→[-π/2,π/2] increasing
• cos⁻¹ [-1,1]→[0,π] decreasing
• tan⁻¹ R→(-π/2,π/2) S-shaped asymptotes ±π/2
• Principal value = angle in principal range
• Example: sin⁻¹(√3/2)=π/3, sin⁻¹(-1/2)=-π/6
3. Domain and Range Table - Must Remember
Domain is input allowed, range is output principal branch. sin⁻¹ domain [-1,1] range [-π/2,π/2], odd function. cos⁻¹ domain [-1,1] range [0,π] decreasing. tan⁻¹ domain R (-∞,∞) range (-π/2,π/2) odd. cot⁻¹ domain R range (0,π). sec⁻¹ domain (-∞,-1]∪[1,∞) range [0,π]-{π/2}. cosec⁻¹ domain same range [-π/2,π/2]-{0}. Remember this table, all questions use it.
Table: sin⁻¹ [-1,1]→[-π/2,π/2], cos⁻¹ [-1,1]→[0,π], tan⁻¹ R→(-π/2,π/2), cot⁻¹ R→(0,π), sec⁻¹ |x|≥1 →[0,π]-{π/2}, cosec⁻¹ |x|≥1 →[-π/2,π/2]-{0}. sin⁻¹,tan⁻¹ odd.
• sin⁻¹: Domain [-1,1] Range [-π/2,π/2] odd
• cos⁻¹: Domain [-1,1] Range [0,π] decreasing
• tan⁻¹: Domain R Range (-π/2,π/2) odd
• cot⁻¹: Domain R Range (0,π)
• sec⁻¹: |x|≥1 →[0,π]-π/2, cosec⁻¹: |x|≥1 →[-π/2,π/2]-0
• Must memorize table
4. Properties of sin⁻¹ and cos⁻¹ - Negative and Complementary
Property 1 Negative: sin⁻¹(-x) = -sin⁻¹(x) odd, cos⁻¹(-x)=π - cos⁻¹(x) not odd, tan⁻¹(-x)=-tan⁻¹(x). Property 2 Complementary: sin⁻¹(x)+cos⁻¹(x)=π/2 for x∈[-1,1]. Because sinθ=x then cos(π/2-θ)=x so cos⁻¹(x)=π/2-θ. Example sin⁻¹(1/2)=π/6, cos⁻¹(1/2)=π/3 sum π/2. Similarly tan⁻¹+cot⁻¹=π/2, sec⁻¹+cosec⁻¹=π/2.
sin⁻¹(-x)=-sin⁻¹ x, cos⁻¹(-x)=π-cos⁻¹ x, tan⁻¹(-x)=-tan⁻¹ x, cot⁻¹(-x)=π-cot⁻¹ x, sec⁻¹(-x)=π-sec⁻¹ x, cosec⁻¹(-x)=-cosec⁻¹ x. Complementary: sin⁻¹ x+cos⁻¹ x=π/2, tan⁻¹ x+cot⁻¹ x=π/2, sec⁻¹ x+cosec⁻¹ x=π/2.
• Negative: sin⁻¹(-x)=-sin⁻¹ x, tan⁻¹(-x)=-tan⁻¹ x odd
• cos⁻¹(-x)=π-cos⁻¹ x, cot⁻¹(-x)=π-cot⁻¹ x
• Complementary: sin⁻¹ x+cos⁻¹ x=π/2
• tan⁻¹ x+cot⁻¹ x=π/2, sec⁻¹ x+cosec⁻¹ x=π/2
• Example: sin⁻¹ 1/2=π/6 cos⁻¹ 1/2=π/3 sum π/2
5. Properties of tan⁻¹, cot⁻¹, sec⁻¹, cosec⁻¹
tan⁻¹ and cot⁻¹ relation: cot⁻¹(x)=tan⁻¹(1/x) for x>0, but for x<0 cot⁻¹(x)=π+tan⁻¹(1/x). sec⁻¹(x)=cos⁻¹(1/x) for |x|≥1, cosec⁻¹(x)=sin⁻¹(1/x). Also sin⁻¹(x)=cos⁻¹(√(1-x²))=tan⁻¹(x/√(1-x²)). These help convert one inverse to another. Example sin⁻¹(3/5)=tan⁻¹(3/4) because triangle with opposite 3 hypotenuse 5 adjacent 4 tan=3/4.
Conversions: sin⁻¹ x = cos⁻¹√(1-x²)=tan⁻¹(x/√(1-x²)), cos⁻¹ x = sin⁻¹√(1-x²)=tan⁻¹(√(1-x²)/x), tan⁻¹ x = sin⁻¹(x/√(1+x²))=cos⁻¹(1/√(1+x²)). Reciprocal: sec⁻¹ x=cos⁻¹(1/x), cosec⁻¹ x=sin⁻¹(1/x), cot⁻¹ x=tan⁻¹(1/x) for x>0.
• cot⁻¹ x = tan⁻¹(1/x) for x>0, = π+tan⁻¹(1/x) for x<0
• sec⁻¹ x = cos⁻¹(1/x), cosec⁻¹ x = sin⁻¹(1/x)
• sin⁻¹ x = tan⁻¹(x/√(1-x²)), cos⁻¹ x = tan⁻¹(√(1-x²)/x)
• Right triangle: sinθ=x ⇒ cosθ=√(1-x²) tanθ=x/√(1-x²)
• Example: sin⁻¹ 3/5 = tan⁻¹ 3/4 (3-4-5 triangle)
6. Sum Formulas - sin⁻¹+cos⁻¹=π/2
Main identity sin⁻¹(x)+cos⁻¹(x)=π/2 proved by right triangle. Take θ=sin⁻¹(x) then sinθ=x, in right triangle opposite x hypotenuse 1 adjacent √(1-x²) so cosθ=√(1-x²) but also cos(π/2-θ)=sinθ=x so cos⁻¹(x)=π/2-θ sum π/2. Similarly tan⁻¹(x)+cot⁻¹(x)=π/2. These are always true where defined. Use to simplify expressions.
sin⁻¹ x+cos⁻¹ x=π/2 for x∈[-1,1], tan⁻¹ x+cot⁻¹ x=π/2 for x∈R, sec⁻¹ x+cosec⁻¹ x=π/2 for |x|≥1. Visual proof with right triangle complementary angles sum 90°=π/2.
• sin⁻¹ x + cos⁻¹ x = π/2 for [-1,1]
• tan⁻¹ x + cot⁻¹ x = π/2 for all R
• sec⁻¹ x + cosec⁻¹ x = π/2 for |x|≥1
• Right triangle proof: complementary angles sum 90°
• Use: Simplify sin⁻¹+cos⁻¹ always π/2
7. Addition Formulas - tan⁻¹ x + tan⁻¹ y
tan⁻¹ x + tan⁻¹ y = tan⁻¹((x+y)/(1-xy)) if xy<1, = π+tan⁻¹((x+y)/(1-xy)) if xy>1 and x,y positive, = -π+... if xy>1 and negative. Example tan⁻¹ 1 + tan⁻¹ 2 = π + tan⁻¹((3)/(1-2))=π+tan⁻¹(-3)=π- tan⁻¹ 3 but principal check gives? Another: tan⁻¹ 1/2 + tan⁻¹ 1/3 = tan⁻¹((5/6)/(1-1/6))=tan⁻¹ 1 =π/4. So two small angles sum to 45°. Very useful for proving.
tan⁻¹ x ± tan⁻¹ y = tan⁻¹((x±y)/(1∓xy)) with conditions xy<1, = π+... if xy>1. Also 2tan⁻¹ x = sin⁻¹(2x/(1+x²))=cos⁻¹((1-x²)/(1+x²))=tan⁻¹(2x/(1-x²)) for |x|<1. Example tan⁻¹ 1/2+tan⁻¹ 1/3=π/4.
• tan⁻¹ x + tan⁻¹ y = tan⁻¹((x+y)/(1-xy)) if xy<1
• If xy>1 x>0 y>0 then = π+tan⁻¹((x+y)/(1-xy))
• tan⁻¹ x - tan⁻¹ y = tan⁻¹((x-y)/(1+xy))
• 2tan⁻¹ x = tan⁻¹(2x/(1-x²)) for |x|<1 = sin⁻¹(2x/(1+x²))
• Example: tan⁻¹ 1/2+tan⁻¹ 1/3=π/4
8. Real Life Usage - Where Inverse Trig Lives
Inverse trig is used to find angle from ratio. Engineer measuring height of building: knows opposite and adjacent, uses tan⁻¹(opposite/adjacent)=angle of elevation. Navigation: GPS uses inverse trig to calculate bearing. Construction: ramp angle uses sin⁻¹(height/length). Physics: projectile motion angle from components uses tan⁻¹(vy/vx). Computer graphics: rotating objects uses atan2. Medical imaging: angle of X-ray.
Real: Height measurement tan⁻¹, navigation bearing, ramp angle sin⁻¹, projectile tan⁻¹(vy/vx), robotics arm angle, computer graphics rotation, architecture slope, astronomy angle of elevation, game development direction.
• Height: tan⁻¹(opposite/adjacent)=angle of elevation
• Navigation: GPS bearing uses tan⁻¹
• Ramp: sin⁻¹(height/length)=inclination
• Projectile: tan⁻¹(vy/vx)=launch angle
• Graphics: atan2 for rotation, games direction
• Engineering: bridge slope, antenna angle
• What: sin θ=x ⇒ θ=sin⁻¹ x, need principal branch to make one-one
• Principal: sin⁻¹ [-1,1]→[-π/2,π/2], cos⁻¹ [-1,1]→[0,π], tan⁻¹ R→(-π/2,π/2)
• Domain Range: sin⁻¹ [-1,1]→[-π/2,π/2], cos⁻¹ [-1,1]→[0,π], tan⁻¹ R→(-π/2,π/2), cot⁻¹ R→(0,π), sec⁻¹ |x|≥1→[0,π]-π/2, cosec⁻¹ |x|≥1→[-π/2,π/2]-0
• Negative: sin⁻¹(-x)=-sin⁻¹ x, tan⁻¹(-x)=-tan⁻¹ x, cos⁻¹(-x)=π-cos⁻¹ x
• Complementary: sin⁻¹+cos⁻¹=π/2, tan⁻¹+cot⁻¹=π/2, sec⁻¹+cosec⁻¹=π/2
• Conversion: sin⁻¹ x = cos⁻¹√(1-x²)=tan⁻¹(x/√(1-x²)), sec⁻¹ x=cos⁻¹(1/x)
• Addition: tan⁻¹ x+tan⁻¹ y=tan⁻¹((x+y)/(1-xy)) xy<1, 2tan⁻¹ x=tan⁻¹(2x/(1-x²))
• Example: tan⁻¹ 1/2+tan⁻¹ 1/3=π/4
• Real: Height angle tan⁻¹, navigation bearing, ramp sin⁻¹, projectile tan⁻¹, graphics atan2