Question

find the six trigonometric function values of the specified angle. sin θ = 12/13 (type an integer or a simplified fraction.) cos θ = 5/13 (type an integer or a simplified fraction.) tan θ = 13/5 (type an integer or a simplified fraction.)

Explanation:

Step1: Recall trig - function definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}$, $\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}$, $\cot\theta=\frac{\text{adjacent}}{\text{opposite}}$. Given opposite side = 12, adjacent side = 5, and hypotenuse = 13.

Step2: Calculate $\sin\theta$

$\sin\theta=\frac{12}{13}$

Step3: Calculate $\cos\theta$

$\cos\theta=\frac{5}{13}$

Step4: Calculate $\tan\theta$

$\tan\theta=\frac{12}{5}$

Step5: Calculate $\csc\theta$

$\csc\theta=\frac{13}{12}$

Step6: Calculate $\sec\theta$

$\sec\theta=\frac{13}{5}$

Step7: Calculate $\cot\theta$

$\cot\theta=\frac{5}{12}$

Answer:

$\sin\theta=\frac{12}{13}$, $\cos\theta=\frac{5}{13}$, $\tan\theta=\frac{12}{5}$, $\csc\theta=\frac{13}{12}$, $\sec\theta=\frac{13}{5}$, $\cot\theta=\frac{5}{12}$