First slide

Trigonometric equations

Question

If $\tan (A - B) = 1 and \sec (A + B) = 2 / \sqrt{3}$ , then the smallest positive values of A and B, respectively, are

Moderate

A

$\frac{25 π}{24}, \frac{19 π}{24}$
B

$\frac{19 π}{24}, \frac{25 π}{24}$
C

$\frac{31 π}{24}, \frac{13 π}{24}$
D

$\frac{13 π}{24}, \frac{31 π}{24}$

Solution

$\tan (A - B) = 1$
$\begin{array}{l} \Rightarrow A - B = n_{1} π + \frac{π}{4} or A - B = \frac{π}{4}, \frac{3 π}{4}, - \frac{3 π}{4}, \dots \\ \sec (A + B) = \frac{2}{\sqrt{3}} \\ \Rightarrow A + B = 2 n_{2} π \pm \frac{π}{6} or A + B = \frac{π}{6}, \frac{11 π}{6}, \dots \end{array}$
For the least positive values of A and B, we have
$\begin{aligned} A + B = \frac{11 π}{6}, A - B = \frac{π}{4} \\ \Rightarrow B = \frac{19 π}{24}, A = \frac{25 π}{24} \end{aligned}$

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