Question

The value of \( 1 + \sec 19^\circ \sin 71^\circ \) is:

• A. \( 2 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 1.5 \)

Hint:

Use complementary angle identities to simplify trigonometric expressions.

Answer 1

The Correct Option is A

Step 1: Use the complementary angle identity.
We know that: \[ \sin 71^\circ = \cos 19^\circ. \] This simplifies the expression to: \[ 1 + \sec 19^\circ \sin 71^\circ = 1 + \sec 19^\circ \cos 19^\circ. \] Step 2: Express \( \sec 19^\circ \).
Since \( \sec \theta = \frac{1}{\cos \theta} \), we can substitute: \[ 1 + \frac{1}{\cos 19^\circ} \cdot \cos 19^\circ = 1 + 1 = 2. \] Step 3: Conclude the result.
Thus, the value of \( 1 + \sec 19^\circ \sin 71^\circ \) is \( 2 \).