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 hypotenuse

By the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 8\sqrt{2}\) and \(b=14\). So \(c=\sqrt{(8\sqrt{2})^{2}+14^{2}}=\sqrt{128 + 196}=\sqrt{324}=18\).

Step2: Calculate sine

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{14}{18}=\frac{7}{9}\)

Step3: Calculate cosine

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8\sqrt{2}}{18}=\frac{4\sqrt{2}}{9}\)

Step4: Calculate tangent

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{14}{8\sqrt{2}}=\frac{7\sqrt{2}}{8}\)

Step5: Calculate cosecant

\(\csc\theta=\frac{1}{\sin\theta}=\frac{9}{7}\)

Step6: Calculate secant

\(\sec\theta=\frac{1}{\cos\theta}=\frac{9}{4\sqrt{2}}=\frac{9\sqrt{2}}{8}\)

Step7: Calculate cotangent

\(\cot\theta=\frac{1}{\tan\theta}=\frac{8}{7\sqrt{2}}=\frac{4\sqrt{2}}{7}\)

Answer:

\(\sin\theta=\frac{7}{9}\), \(\cos\theta=\frac{4\sqrt{2}}{9}\), \(\tan\theta=\frac{7\sqrt{2}}{8}\), \(\csc\theta=\frac{9}{7}\), \(\sec\theta=\frac{9\sqrt{2}}{8}\), \(\cot\theta=\frac{4\sqrt{2}}{7}\)