Question

In Exercises \(37-40\) , use the given information to find the values of the six trigonometric functions at the angle \(\theta\) . Give exact answers. $$\theta=\sin ^{-1}\left(\frac{8}{17}\right)$$

Short answer

The six trigonometric functions values are: \( \sin(\theta) = \frac{8}{17} \), \( \cos(\theta) = \frac{15}{17} \), \( \tan(\theta) = \frac{8}{15} \), \( \cot(\theta) = \frac{15}{8} \), \( \csc(\theta) = \frac{17}{8} \), and \( \sec(\theta) = \frac{17}{15} \)

Step-by-step solution

Identify Known Values

Given \( \sin(\theta) = \frac{8}{17} \), this means in a right triangle with angle \( \theta \), the length of the opposite side is 8 and the hypotenuse is 17.

Find the adjacent side

We can use the Pythagorean theorem \( a^2 + b^2 = c^2 \) where the hypotenuse \( c = 17 \) and one side \( a = 8 \). Solving for \( b \), which represents the adjacent side in our case: \( b = \sqrt{c^2 - a^2} = \sqrt{17^2 - 8^2} =15. \)

Find Cosine value

We can use the formula \( \cos(\theta) = \frac{adjacent}{hypotenuse} = \frac{15}{17} \).

Find Tangent value

We can use the formula \( \tan(\theta) = \frac{opposite}{adjacent} = \frac{8}{15} \) to find the tangent of the angle.

Finding the remaining trigonometric functions

Finally, using the identities: \(\cot(\theta) = \frac{1}{\tan(\theta)}\), yielding \(\cot(\theta) = \frac{15}{8}; \csc(\theta) = \frac{1}{\sin(\theta)}\), which gives us \(\csc(\theta) = \frac{17}{8}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\) providing us with \(\sec(\theta) = \frac{17}{15}\).