Open Math
Inverse trigonometric functions Trigonometric functions · ME-12-02
Inverse trig functions Q1 Straightforward
Find the exact value of tan − 1 ( 1 ) \tan^{-1}(1) tan − 1 ( 1 ) in radians. Give your answer as a decimal to 4 decimal places.
Q2 Straightforward
Find the exact value of sin − 1 ( 0 ) \sin^{-1}(0) sin − 1 ( 0 ) in radians.
Q3 Straightforward
Find the exact value of cos − 1 ( 1 ) \cos^{-1}(1) cos − 1 ( 1 ) in radians.
Q4 Straightforward
Find the exact value of sin − 1 ( 1 ) \sin^{-1}(1) sin − 1 ( 1 ) in radians. Give your answer as a decimal to 4 decimal places.
Q5 Straightforward
The function sin − 1 ( x ) \sin^{-1}(x) sin − 1 ( x ) is defined only for values of x x x in what interval? A. [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] B. ( − ∞ , ∞ ) (-\infty, \infty) ( − ∞ , ∞ ) C. [ − π 2 , π 2 ] \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] [ − 2 π , 2 π ] D. [ 0 , π ] [0, \pi] [ 0 , π ]
Q6 Moderate
Find the exact value of cos − 1 ( − 1 2 ) \cos^{-1}\!\left(-\dfrac{1}{2}\right) cos − 1 ( − 2 1 ) in radians. Give your answer as a decimal to 4 decimal places.
Q7 Moderate
Find the exact value of sin − 1 ( − 2 2 ) \sin^{-1}\!\left(-\dfrac{\sqrt{2}}{2}\right) sin − 1 ( − 2 2 ) in radians. Give your answer as a decimal to 4 decimal places.
Q8 Moderate
Find the exact value of tan − 1 ( − 3 ) \tan^{-1}(-\sqrt{3}) tan − 1 ( − 3 ) in radians. Give your answer as a decimal to 4 decimal places.
Q9 Moderate
Solve sin − 1 ( x ) = π 3 \sin^{-1}(x) = \dfrac{\pi}{3} sin − 1 ( x ) = 3 π for x x x . Give your answer as a decimal to 4 decimal places.
Q10 Moderate
What is the range of tan − 1 ( x ) \tan^{-1}(x) tan − 1 ( x ) ? A. ( − π 2 , π 2 ) \left(-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right) ( − 2 π , 2 π ) (open interval, not including the endpoints) B. [ − π 2 , π 2 ] \left[-\dfrac{\pi}{2},\, \dfrac{\pi}{2}\right] [ − 2 π , 2 π ] (closed interval, including the endpoints) C. ( − ∞ , ∞ ) (-\infty, \infty) ( − ∞ , ∞ ) D. [ 0 , π ] [0, \pi] [ 0 , π ]
Q11 Challenging
Find the exact value of cos ( sin − 1 ( 3 5 ) ) \cos\!\left(\sin^{-1}\!\left(\dfrac{3}{5}\right)\right) cos ( sin − 1 ( 5 3 ) ) .
Q12 Challenging
Find the exact value of sin − 1 ( sin ( 5 π 6 ) ) \sin^{-1}\!\left(\sin\!\left(\dfrac{5\pi}{6}\right)\right) sin − 1 ( sin ( 6 5 π ) ) in radians. Give your answer as a decimal to 4 decimal places.
Q13 Challenging
Solve 2 sin − 1 ( x ) = π 2 2\sin^{-1}(x) = \dfrac{\pi}{2} 2 sin − 1 ( x ) = 2 π for x x x . Give your answer as a decimal to 4 decimal places.
Properties and calculus Q14 Straightforward
The derivative of sin − 1 ( x ) \sin^{-1}(x) sin − 1 ( x ) is d d x sin − 1 ( x ) = 1 1 − x 2 \dfrac{d}{dx}\sin^{-1}(x) = \dfrac{1}{\sqrt{1-x^2}} d x d sin − 1 ( x ) = 1 − x 2 1 . Evaluate this derivative at x = 0 x = 0 x = 0 .
Q15 Straightforward
The derivative of tan − 1 ( x ) \tan^{-1}(x) tan − 1 ( x ) is d d x tan − 1 ( x ) = 1 1 + x 2 \dfrac{d}{dx}\tan^{-1}(x) = \dfrac{1}{1 + x^2} d x d tan − 1 ( x ) = 1 + x 2 1 . Evaluate this derivative at x = 1 x = 1 x = 1 .
Q16 Straightforward
Differentiate f ( x ) = tan − 1 ( 2 x ) f(x) = \tan^{-1}(2x) f ( x ) = tan − 1 ( 2 x ) using the chain rule. Find f ′ ( 0 ) f'(0) f ′ ( 0 ) .
Q17 Moderate
Differentiate y = cos − 1 ( 1 − x 2 ) y = \cos^{-1}(1 - x^2) y = cos − 1 ( 1 − x 2 ) and find d y d x \dfrac{dy}{dx} d x d y at x = 1 x = 1 x = 1 .
Q18 Moderate
Evaluate ∫ 0 1 / 2 1 1 − x 2 d x \displaystyle\int_0^{1/2} \frac{1}{\sqrt{1 - x^2}}\,dx ∫ 0 1/2 1 − x 2 1 d x . Give your answer in radians to 4 decimal places.
Q19 Moderate
Evaluate ∫ 0 1 1 1 + x 2 d x \displaystyle\int_0^1 \frac{1}{1 + x^2}\,dx ∫ 0 1 1 + x 2 1 d x . Give your answer in radians to 4 decimal places.
Q20 Moderate
Evaluate ∫ 0 3 1 1 + x 2 d x \displaystyle\int_0^{\sqrt{3}} \frac{1}{1 + x^2}\,dx ∫ 0 3 1 + x 2 1 d x . Give your answer in radians to 4 decimal places.
Q21 Challenging
Differentiate y = sin − 1 ( x 3 ) y = \sin^{-1}\!\left(\dfrac{x}{3}\right) y = sin − 1 ( 3 x ) and find the exact value of d y d x \dfrac{dy}{dx} d x d y at x = 0 x = 0 x = 0 . Express your answer as a fraction p / q p/q p / q in lowest terms.
Q22 Challenging
Evaluate ∫ 0 1 1 4 − x 2 d x \displaystyle\int_0^1 \frac{1}{\sqrt{4 - x^2}}\,dx ∫ 0 1 4 − x 2 1 d x . Give your answer in radians to 4 decimal places. **Hint:** Use ∫ 1 a 2 − x 2 d x = sin − 1 ( x a ) + C \displaystyle\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \sin^{-1}\!\left(\frac{x}{a}\right) + C ∫ a 2 − x 2 1 d x = sin − 1 ( a x ) + C .
Q23 Challenging
Evaluate ∫ 0 2 1 4 + x 2 d x \displaystyle\int_0^2 \frac{1}{4 + x^2}\,dx ∫ 0 2 4 + x 2 1 d x . Give your answer in radians to 4 decimal places. **Hint:** Use ∫ 1 a 2 + x 2 d x = 1 a tan − 1 ( x a ) + C \displaystyle\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\tan^{-1}\!\left(\frac{x}{a}\right) + C ∫ a 2 + x 2 1 d x = a 1 tan − 1 ( a x ) + C .
Q24 Challenging
Differentiate f ( x ) = x tan − 1 ( x ) f(x) = x\tan^{-1}(x) f ( x ) = x tan − 1 ( x ) and find f ′ ( 1 ) f'(1) f ′ ( 1 ) . Give your answer to 4 decimal places.
Q25 Challenging
Differentiate y = cos − 1 ( 1 x ) y = \cos^{-1}\!\left(\dfrac{1}{x}\right) y = cos − 1 ( x 1 ) for x > 1 x > 1 x > 1 and find d y d x \dfrac{dy}{dx} d x d y at x = 2 x = 2 x = 2 . Give your answer to 4 decimal places.
Worked solutions and answers at openmath.au/year-12/extension-1/inverse-trigonometric-functions