Question

use the information contained in the figure to determine the values of the six trigonometric functions of θ. write the exact answers and simplify. do not round.

Explanation:

Step1: Find the adjacent side

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 65\) and \(b = 56\). Let the adjacent side be \(a\). Then \(a=\sqrt{c^{2}-b^{2}}=\sqrt{65^{2}-56^{2}}=\sqrt{(65 + 56)(65 - 56)}=\sqrt{121\times9}=\sqrt{1089}=33\).

Step2: Calculate sine

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{56}{65}\)

Step3: Calculate cosine

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{33}{65}\)

Step4: Calculate tangent

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{56}{33}\)

Step5: Calculate cosecant

\(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{65}{56}\)

Step6: Calculate secant

\(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{65}{33}\)

Step7: Calculate cotangent

\(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{33}{56}\)

Answer:

\(\sin\theta=\frac{56}{65}\), \(\cos\theta=\frac{33}{65}\), \(\tan\theta=\frac{56}{33}\), \(\csc\theta=\frac{65}{56}\), \(\sec\theta=\frac{65}{33}\), \(\cot\theta=\frac{33}{56}\)