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. $$\theta=\tan ^{-1}\left(-\frac{5}{12}\right)$$

Short answer

The values of the six trigonometric functions at the angle \( \theta = \tan^{-1}(-5/12) \) are: \( \sin(\theta) = -5/13 \), \( \cos(\theta) = 12/13 \), \( \tan(\theta) = -5/12 \), \( \csc(\theta) = -13/5 \), \( \sec(\theta) = 13/12 \), and \( \cot(\theta) = -12/5 \).

Step-by-step solution

Determine the hypotenuse

Using the Pythagorean theorem for the right triangle, the hypotenuse \( r \) is \( \sqrt{(12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \) .

Compute sine value

Now, the sine of the angle \( \theta \), \( \sin(\theta) \), is determined by the ratio of the opposite side to the hypotenuse: \( \sin(\theta) = \frac{-5}{13} \) .

Compute cosine value

The cosine of the angle \( \theta \), \( \cos(\theta) \), is determined by the ratio of the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{12}{13} \).

Compute tangent value

The tangent of the angle \( \theta \), \( \tan(\theta) \), is determined by the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{-5}{12} \), which is given already.

Compute the remaining trigonometric functions

For the remaining functions, cosecant \( \csc(\theta) \) is the reciprocal of sin(\( \theta \)), secant \( \sec(\theta) \) is the reciprocal of cos(\( \theta \)), and cotangent \( \cot(\theta) \) is the reciprocal of tan(\( \theta \)). So, \( \csc(\theta) = \frac{-13}{5} \), \( \sec(\theta) = \frac{13}{12} \), \( \cot(\theta) = \frac{-12}{5} \).