Question

Which of the following is true?

• A. \( \sin(A + B) = \sin A + \sin B \)
• B. The value of \( \sin \theta \) increases as \( \theta \) increases, \( 0^\circ \leq \theta \leq 90^\circ \)
• C. The value of \( \cos \theta \) increases as \( \theta \) increases, \( 0^\circ \leq \theta \leq 90^\circ \)
• D. \( \sin \theta = \cos \theta \) for all values of \( \theta \)

Hint:

To determine the behavior of trigonometric functions, refer to their graphs between \( 0^\circ \) and \( 90^\circ \). - \( \sin \theta \) increases - \( \cos \theta \) decreases - \( \sin \theta = \cos \theta \) only at \( \theta = 45^\circ \)

Answer 1

The Correct Option is B

Step 1: Analyze Option (1) 
Recall the identity: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \] So, Option (1) is incorrect because it falsely states that: \[ \sin(A + B) = \sin A + \sin B \] 
Step 2: Analyze Option (2) 
From the unit circle or sine graph: \[ \text{For } 0^\circ \leq \theta \leq 90^\circ, \sin \theta \text{ increases from } 0 \text{ to } 1 \] Hence, Option (2) is true. 
Step 3: Analyze Option (3) 
For \( 0^\circ \leq \theta \leq 90^\circ \), the value of \( \cos \theta \) actually decreases: \[ \cos(0^\circ) = 1,\quad \cos(90^\circ) = 0 \] So, Option (3) is false. 
Step 4: Analyze Option (4) 
\[ \sin \theta = \cos \theta \text{ only when } \theta = 45^\circ \] This is not true for all values of \( \theta \), so Option (4) is false.