Inverse Trigonometric Functions — Practice

Question 1

Find the principal value of sin^{-1}(1).

Accepted Answer

sin(pi/2)=1, and pi/2 is inside the principal branch [-pi/2,pi/2]. Answer: Principal value is pi/2.

Question 2

Find the principal value of cos^{-1}(-1).

Accepted Answer

cos(pi)=-1, and pi is inside the principal branch [0,pi]. Answer: Principal value is pi.

Question 3

Find the value of cos^{-1}(cos(4pi/3)).

Accepted Answer

4pi/3 is not in [0,pi]. cos(4pi/3)=cos(2pi-4pi/3)=cos(2pi/3), and 2pi/3 is in [0,pi]. Answer: cos^{-1}(cos(4pi/3)) = 2pi/3.

Question 4

Simplify tan^{-1}(x/sqrt(a^2-x^2)) for |x|<a, using the substitution x=a sinθ.

Accepted Answer

With x=a sinθ, sqrt(a^2-x^2)=a cosθ, so the expression becomes tan^{-1}(tanθ)=θ. Answer: Since sinθ=x/a, the simplified form is sin^{-1}(x/a).

Question 5

Find the value of tan^{-1}(sqrt3) - sec^{-1}(-2).

Accepted Answer

tan^{-1}(sqrt3)=pi/3. sec^{-1}(-2): need cosθ=-1/2 with θ in [0,pi]-{pi/2}, giving θ=2pi/3. Answer: pi/3 - 2pi/3 = -pi/3.

Question 6

Prove that 2 tan^{-1}(1/2) = sin^{-1}(4/5).

Accepted Answer

Let tan^{-1}(1/2)=θ, so tanθ=1/2, sinθ=1/sqrt5, cosθ=2/sqrt5. sin2θ=2sinθcosθ=2(1/sqrt5)(2/sqrt5)=4/5. Since θ is small and positive, 2θ is within [-pi/2,pi/2]. Answer: sin(2θ)=4/5, so 2θ=sin^{-1}(4/5), proving the identity.

Question 7

Solve tan^{-1}(2x) + tan^{-1}(3x) = pi/4 for x, and verify the root is not extraneous.

Accepted Answer

Using the addition formula: tan^{-1}(2x)+tan^{-1}(3x)=tan^{-1}((2x+3x)/(1-6x^2))=pi/4, so (5x)/(1-6x^2)=1, giving 5x=1-6x^2, so 6x^2+5x-1=0, factoring (6x-1)(x+1)=0, so x=1/6 or x=-1. Checking x=-1: tan^{-1}(-2)+tan^{-1}(-3) is negative, not pi/4 — extraneous. Checking x=1/6: both tan^{-1}(1/3) and tan^{-1}(1/2) are small positive angles summing plausibly to pi/4. Answer: x = 1/6.

Question 8

Find the value of sin^{-1}(sin(5pi/6)).

Accepted Answer

5pi/6 is not in [-pi/2,pi/2]. sin(5pi/6)=sin(pi-5pi/6)=sin(pi/6), and pi/6 is in [-pi/2,pi/2]. Answer: sin^{-1}(sin(5pi/6)) = pi/6.

Practice — Inverse Trigonometric Functions

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