Questions: Let θ be an angle such that tan θ = 4/7 and cos θ > 0. Find the exact values of sec θ and sin θ. sec θ = sin θ =

Solution Steps

To find the exact values of \(\sec \theta\) and \(\sin \theta\) given that \(\tan \theta = \frac{4}{7}\) and \(\cos \theta > 0\), we can use the following approach:

1. Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) to express \(\sin \theta\) and \(\cos \theta\) in terms of a right triangle with opposite side 4 and adjacent side 7.
2. Calculate the hypotenuse using the Pythagorean theorem: \(\text{hypotenuse} = \sqrt{4^2 + 7^2}\).
3. Use the hypotenuse to find \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\) and \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
4. Calculate \(\sec \theta\) as the reciprocal of \(\cos \theta\).

Given the opposite side \(4\) and the adjacent side \(7\), we can find the hypotenuse \(h\) using the Pythagorean theorem: \[ h = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65} \approx 8.0623 \]

Using the definition of sine, we have: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{\sqrt{65}} \approx 0.4961 \]

Using the definition of cosine, we find: \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{\sqrt{65}} \approx 0.8682 \]

The secant function is the reciprocal of cosine: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{65}}{7} \approx 1.1518 \]

Final Answer