Question

If \(\sec(7^\circ - 2\alpha) = \csc(5\alpha - 7^\circ)\), then the value of \(\alpha\) is:

• A. $ 60^\circ $
• B. $ 50^\circ $
• C. $ 40^\circ $
• D. $ 30^\circ $

Hint:

When dealing with trigonometric equations involving secant and cosecant, use the complementary angle relationships to simplify the problem. Equating the arguments of trigonometric functions often leads to straightforward algebraic solutions.

Answer 1

The Correct Option is D

Step 1: Use the Relationship Between Secant and Cosecant. Recall that: \[ \sec \theta = \csc(90^\circ - \theta) \] Thus, the given equation: \[ \sec(7^\circ - 2\alpha) = \csc(5\alpha - 7^\circ) \] can be rewritten using the relationship above: \[ \csc(90^\circ - (7^\circ - 2\alpha)) = \csc(5\alpha - 7^\circ) \] Step 2: Simplify the Argument of the Cosecant Function. Simplify \( 90^\circ - (7^\circ - 2\alpha) \): \[ 90^\circ - (7^\circ - 2\alpha) = 90^\circ - 7^\circ + 2\alpha = 83^\circ + 2\alpha \] Thus, the equation becomes: \[ \csc(83^\circ + 2\alpha) = \csc(5\alpha - 7^\circ) \] Step 3: Equate the Arguments. For the cosecant functions to be equal, their arguments must differ by integer multiples of \( 360^\circ \), or be directly equal. We assume: \[ 83^\circ + 2\alpha = 5\alpha - 7^\circ \] Step 4: Solve for \( \alpha \). Rearranging the equation: Start with the equation: \[ 83^\circ + 2\alpha = 5\alpha - 7^\circ \] Move all terms to one side: \[ 83^\circ + 2\alpha - 5\alpha + 7^\circ = 0 \] Combine like terms: \[ (83^\circ + 7^\circ) - 3\alpha = 0 \] \[ 90^\circ = 3\alpha \] Now solve for \( \alpha \): \[ \alpha = \frac{90^\circ}{3} = 30^\circ \] Step 5: Analyze the Options. Option (1): \( 60^\circ \) — Incorrect
Option (2): \( 50^\circ \) — Incorrect
Option (3): \( 40^\circ \) — Incorrect
Option (4): \( 30^\circ \) — Correct Step 6: Final Answer. \[ (4) \quad \mathbf{30^\circ} \]