Question

find cosθ, secθ, and cotθ, where θ is the angle shown in the figure. give exact values, not decimal approximations. cosθ = secθ = cotθ =

Explanation:

Step1: Find the adjacent side

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Let the adjacent side be \(a\), \(c = 25\), \(b=24\). Then \(a=\sqrt{25^{2}-24^{2}}=\sqrt{(25 + 24)(25 - 24)}=\sqrt{49}=7\)

Step2: Calculate \(\cos\theta\)

By the definition of cosine in a right - triangle \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). So \(\cos\theta=\frac{7}{25}\)

Step3: Calculate \(\sec\theta\)

Since \(\sec\theta=\frac{1}{\cos\theta}\), then \(\sec\theta=\frac{25}{7}\)

Step4: Calculate \(\cot\theta\)

By the definition of cotangent \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\). So \(\cot\theta=\frac{7}{24}\)

Answer:

\(\cos\theta=\frac{7}{25}\), \(\sec\theta=\frac{25}{7}\), \(\cot\theta=\frac{7}{24}\)