Question

If \( \cos \theta = \frac{4}{5} \), find the value of \( \sin \theta \cos \theta + \tan^2 \theta \).

Hint:

To solve for trigonometric expressions involving multiple functions, use the Pythagorean identity and trigonometric ratios to simplify the calculations.

Answer 1

We are given that: \[ \cos \theta = \frac{4}{5}. \] We can use the Pythagorean identity to find \( \sin \theta \). The identity is: \[ \sin^2 \theta + \cos^2 \theta = 1. \] Substitute \( \cos \theta = \frac{4}{5} \) into this equation: \[ \sin^2 \theta + \left( \frac{4}{5} \right)^2 = 1, \] \[ \sin^2 \theta + \frac{16}{25} = 1 \quad \Rightarrow \quad \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}. \] Thus: \[ \sin \theta = \frac{3}{5}. \] Now, we need to find \( \sin \theta \cos \theta + \tan^2 \theta \). First, calculate \( \sin \theta \cos \theta \): \[ \sin \theta \cos \theta = \frac{3}{5} \times \frac{4}{5} = \frac{12}{25}. \] Next, calculate \( \tan \theta \). We know that: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}. \] Now, calculate \( \tan^2 \theta \): \[ \tan^2 \theta = \left( \frac{3}{4} \right)^2 = \frac{9}{16}. \] Finally, add \( \sin \theta \cos \theta \) and \( \tan^2 \theta \): \[ \sin \theta \cos \theta + \tan^2 \theta = \frac{12}{25} + \frac{9}{16}. \] To add these fractions, find a common denominator. The least common denominator of 25 and 16 is 400. Rewrite the fractions: \[ \frac{12}{25} = \frac{192}{400}, \quad \frac{9}{16} = \frac{225}{400}. \] Now, add the fractions: \[ \frac{192}{400} + \frac{225}{400} = \frac{417}{400}. \]
Conclusion: The value of \( \sin \theta \cos \theta + \tan^2 \theta \) is \( \frac{417}{400} \).