7.1/2Inverse Trig Functions

Front

What is the definition of inverse trigonometric functions?

Back

Inverse trigonometric functions are functions that reverse the action of the standard trigonometric functions. They take a ratio and return an angle. Common inverse functions include \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \).

Front

What is the range of the function \( \sin^{-1}(x) \)?

Back

The range of \( \sin^{-1}(x) \) is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

Front

What is the range of the function \( \cos^{-1}(x) \)?

Back

The range of \( \cos^{-1}(x) \) is \( [0, \pi] \).

Front

What is the range of the function \( \tan^{-1}(x) \)?

Back

The range of \( \tan^{-1}(x) \) is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

Front

How do you find \( \tan(\sin^{-1}(x)) \)?

Back

To find \( \tan(\sin^{-1}(x)) \), draw a right triangle where the opposite side is \( x \) and the hypotenuse is \( 1 \). Use the Pythagorean theorem to find the adjacent side, then calculate \( \tan \) as \( \frac{\text{opposite}}{\text{adjacent}} \).

Front

What is the relationship between \( \sin \) and \( \cos \) for complementary angles?

Back

For complementary angles, \( \sin(\theta) = \cos(90^{\circ} - \theta) \) or \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \).

Front

What is the value of \( \tan(\frac{\pi}{4}) \)?

Back

The value of \( \tan(\frac{\pi}{4}) \) is 1.

Front

How do you approximate \( \sec^{-1}(x) \) using a calculator?

Back

To approximate \( \sec^{-1}(x) \), use the calculator to find \( \cos^{-1}(\frac{1}{x}) \) for \( x > 1 \) or \( x < -1 \).

Front

What is the exact value of \( \cos^{-1}(\sin(\frac{\pi}{4})) \)?

Back

The exact value of \( \cos^{-1}(\sin(\frac{\pi}{4})) \) is \( \frac{\pi}{4} \).

Front

What is the value of \( \sin(\tan^{-1}(1)) \)?

Back

The value of \( \sin(\tan^{-1}(1)) \) is \( \frac{\sqrt{2}}{2} \).