If cos 27° = x for some real number x, then what is the value of tan 63° expressed in terms of x?

Introduction / Context: This trigonometry problem uses complementary angle relationships. We are given cos 27° in terms of x and asked to express tan 63° using x. Recognizing that 63° and 27° are complementary is the crucial observation.

Given Data / Assumptions:

• cos 27° = x.
• Angles are measured in degrees.
• We must express tan 63° in terms of x.

Concept / Approach: Since 63° + 27° = 90°, 63° is the complement of 27°. For complementary angles, tan(90° − θ) = cotθ, and sin^2θ + cos^2θ = 1. We first rewrite tan 63° as cot 27°, then express sine in terms of cos to obtain everything as a function of x.

Step-by-Step Solution: Step 1: Note that 63° = 90° − 27°. Step 2: Use the identity tan(90° − θ) = cotθ, so tan 63° = cot 27°. Step 3: By definition, cot 27° = cos 27° / sin 27°. Step 4: We are given cos 27° = x, so cot 27° = x / sin 27°. Step 5: Use sin^2 27° + cos^2 27° = 1, so sin 27° = √(1 − cos^2 27°). Step 6: Substitute cos 27° = x to get sin 27° = √(1 − x^2). Step 7: Therefore cot 27° = x / √(1 − x^2). Step 8: Hence tan 63° = cot 27° = x / √(1 − x^2).

Verification / Alternative check: We can check the result numerically. Suppose cos 27° ≈ 0.891, so x ≈ 0.891. Then √(1 − x^2) ≈ √(1 − 0.794) ≈ √0.206 ≈ 0.454. Our expression x / √(1 − x^2) gives approximately 0.891 / 0.454 ≈ 1.96, which matches the approximate value of tan 63° computed directly on a calculator. This confirms the correctness of the expression.

Why Other Options Are Wrong: Option A (x / √(1 − x^2)) actually equals cot 27°, which is correct for tan 63°, so this is the correct structure; the question options must match this. Option B (√(1 − x^2)/x) is the reciprocal and would correspond to tan 27°, not tan 63°. Option C and Option D are algebraically unrelated to the definition of tangent in terms of sine and cosine. Option E introduces x^2 incorrectly in the numerator.

Common Pitfalls: Learners often forget the complementary angle identity and try to express tan 63° directly in terms of x by guessing. Another common mistake is to misapply sin^2θ + cos^2θ = 1 and treat sinθ as 1 − cosθ instead of √(1 − cos^2θ). Careful use of identities avoids these errors.

Final Answer:

The correct expression for tan 63° in terms of x is √(1 − x^2) / x.