Question

If \(sin(A-B)=\frac 12\) and \(cos(A+B)=\frac 12\), then \(∠A, ∠B\) ?

• A. 45°, 15°
• B. 15°, 45°
• C. 45°, 30°
• D. 30°, 15°

Answer 1

The Correct Option is A

To solve the problem, we are given:

• $ \sin(A - B) = \frac{1}{2} $
• $ \cos(A + B) = \frac{1}{2} $

1. Using Standard Trigonometric Values:
We know that:

• $ \sin(30^\circ) = \frac{1}{2} $
• $ \cos(60^\circ) = \frac{1}{2} $

This implies:

• $ A - B = 30^\circ $
• $ A + B = 60^\circ $

 

2. Solving the System of Equations:
Add the two equations:
$ (A - B) + (A + B) = 30^\circ + 60^\circ $
$ 2A = 90^\circ \Rightarrow A = 45^\circ $

Now substitute $A = 45^\circ$ into $A - B = 30^\circ$:
$ 45^\circ - B = 30^\circ \Rightarrow B = 15^\circ $

Final Answer:
$ \angle A = 45^\circ, \angle B = 15^\circ \Rightarrow {45^\circ, 15^\circ} $