First slide

Trigonometric Identities

Question

If $\sin θ + \cos θ = \frac{1}{5} and 0 \leq θ < π$ then $\tan θ$ is

Moderate

A

-4/3
B

-¾
C

¾
D

4/3

Solution

$\begin{array}{l} \cos θ + \sin θ = \frac{1}{5} \\ \Rightarrow 1 + \tan θ = \frac{1}{5} \sec θ (dividing by \cos θ) \\ \Rightarrow 25 + 50 \tan θ + 25 \tan^{2} θ = \sec^{2} θ \\ or 12 \tan^{2} θ + 25 \tan θ + 12 = 0 \\ or 12 \tan^{2} θ + 16 \tan θ + 9 \tan θ + 12 = 0 \\ or (4 \tan θ + 3) (3 \tan θ + 4) = 0 \\ \Rightarrow \tan θ = - \frac{3}{4} or \tan θ = - \frac{4}{3} \\ \tan θ = - \frac{3}{4} is rejected \end{array}$

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