Question

use the pythagorean theorem to find the length of the missing side of the right triangle. then find the value of each of the six trigonometric functions of θ. complete the table by using the names of the sides to express each trigonometric function as a ratio. sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =

Explanation:

Step1: Find the length of the hypotenuse c

By the Pythagorean Theorem \(c^{2}=a^{2}+b^{2}\), where \(a = 12\) and \(b=16\). So \(c=\sqrt{12^{2}+16^{2}}=\sqrt{144 + 256}=\sqrt{400}=20\).

Step2: Define the six - trigonometric functions

• \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}=\frac{12}{20}=\frac{3}{5}\)
• \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}=\frac{16}{20}=\frac{4}{5}\)
• \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{12}{16}=\frac{3}{4}\)
• \(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{c}{a}=\frac{20}{12}=\frac{5}{3}\)
• \(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{c}{b}=\frac{20}{16}=\frac{5}{4}\)
• \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}=\frac{b}{a}=\frac{16}{12}=\frac{4}{3}\)

Answer:

\(\sin\theta=\frac{3}{5}\), \(\cos\theta=\frac{4}{5}\), \(\tan\theta=\frac{3}{4}\), \(\csc\theta=\frac{5}{3}\), \(\sec\theta=\frac{5}{4}\), \(\cot\theta=\frac{4}{3}\)