First slide

Trigonometric equations

Question

If $T a n θ + S e c θ = \sqrt{3}$ , then the principal value of $(θ + \frac{π}{6})$ is

Moderate

A

$\frac{π}{3}$
B

$\frac{π}{4}$
C

$\frac{2 π}{3}$
D

$\frac{π}{2}$

Solution

$\tan θ + \sec θ = \sqrt{3} \Rightarrow \sin θ + 1 = \sqrt{3} \cos θ$
$\Rightarrow \sqrt{3} \cos θ - \sin θ = 1 \Rightarrow \frac{\sqrt{3}}{2} \cos θ - \frac{1}{2} \sin θ = \frac{1}{2}$
$\Rightarrow \cos θ \cos \frac{π}{6} - \sin θ \sin \frac{π}{6} = \frac{1}{2} \Rightarrow \cos (θ + \frac{π}{6}) = \frac{1}{2}$
$∴$ PS of $θ + \frac{π}{6} = \frac{π}{3}$

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