Question

find the six trigonometric function values of the specified angle. (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find hypotenuse

Use Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 5\) and \(b=7\). So \(c=\sqrt{5^{2}+7^{2}}=\sqrt{25 + 49}=\sqrt{74}\).

Step2: Calculate sine

\(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{5}{\sqrt{74}}=\frac{5\sqrt{74}}{74}\)

Step3: Calculate cosine

\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{7}{\sqrt{74}}=\frac{7\sqrt{74}}{74}\)

Step4: Calculate tangent

\(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{5}{7}\)

Step5: Calculate cosecant

\(\csc(\theta)=\frac{1}{\sin(\theta)}=\frac{\sqrt{74}}{5}\)

Step6: Calculate secant

\(\sec(\theta)=\frac{1}{\cos(\theta)}=\frac{\sqrt{74}}{7}\)

Step7: Calculate cotangent

\(\cot(\theta)=\frac{1}{\tan(\theta)}=\frac{7}{5}\)

Answer:

\(\sin(\theta)=\frac{5\sqrt{74}}{74}\), \(\cos(\theta)=\frac{7\sqrt{74}}{74}\), \(\tan(\theta)=\frac{5}{7}\), \(\csc(\theta)=\frac{\sqrt{74}}{5}\), \(\sec(\theta)=\frac{\sqrt{74}}{7}\), \(\cot(\theta)=\frac{7}{5}\)