Questions: Let (8,6) be a point on the terminal side of θ. Find the exact values of cos θ, sec θ, and cot θ. cos θ =□ sec θ =□ cot θ =□

Solution Steps

To find the trigonometric values for the given point \((8,6)\) on the terminal side of \(\theta\), we need to:

1. Calculate the hypotenuse \(r\) using the Pythagorean theorem.
2. Use the definitions of cosine, secant, and cotangent to find the required values.

Given the point \((8, 6)\), we can find the hypotenuse \(r\) using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10. \]

Using the definition of cosine: \[ \cos \theta = \frac{x}{r} = \frac{8}{10} = 0.8. \]

The secant is the reciprocal of cosine: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{0.8} = 1.25. \]

Using the definition of cotangent: \[ \cot \theta = \frac{x}{y} = \frac{8}{6} = \frac{4}{3} \approx 1.3333. \]

Final Answer