Given cos θ = 5/13 and 3π/2 ≤ θ ≤ 2π, find -8tan θ + tan² θ - 16.

Steps to Solve

1. Identify the given information

We have $ \cos θ = \frac{5}{13} $ and $ \frac{3\pi}{2} \leq θ \leq 2\pi $.

2. Find sin θ using the Pythagorean identity

Using the identity $ \sin^2 θ + \cos^2 θ = 1 $, we can find $ \sin θ $: $$ \sin^2 θ = 1 - \cos^2 θ $$ Substituting the given cosine value: $$ \sin^2 θ = 1 - \left( \frac{5}{13} \right)^2 $$ Calculate: $$ \sin^2 θ = 1 - \frac{25}{169} = \frac{144}{169} $$ Thus, $ \sin θ = -\sqrt{\frac{144}{169}} = -\frac{12}{13} $ (since $\sin θ$ is negative in the fourth quadrant).

3. Find tan θ

Using the definition of tangent: $$ \tan θ = \frac{\sin θ}{\cos θ} $$ Substituting the values we have: $$ \tan θ = \frac{-\frac{12}{13}}{\frac{5}{13}} = -\frac{12}{5} $$

4. Calculate the expression -8tan θ + tan² θ - 16

We substitute $ \tan θ $ into the expression: $$ -8\tan θ + \tan^2 θ - 16 $$ Calculating each term:

• First term: $ -8\tan θ = -8 \left( -\frac{12}{5} \right) = \frac{96}{5} $
• Second term: $ \tan^2 θ = \left( -\frac{12}{5} \right)^2 = \frac{144}{25} $
• Constant term: $ -16 = -\frac{80}{5} = -\frac{400}{25} $

Combining all terms: $$ \frac{96}{5} + \frac{144}{25} - \frac{400}{25} $$

5. Convert terms to have a common denominator

The common denominator is 25: $$ \frac{96 \cdot 5}{25} + \frac{144}{25} - \frac{400}{25} = \frac{480}{25} + \frac{144}{25} - \frac{400}{25} $$

6. Combine the fractions

Combine the numerators: $$ \frac{480 + 144 - 400}{25} = \frac{224}{25} $$