Difficulty: Medium
Correct Answer: x = √[(1 + cos 2A)/2]
Explanation:
Introduction / Context:
This problem checks understanding of trigonometric half angle identities, specifically the relationship between cos A and cos 2A. Knowing how to rewrite cosine of an angle in terms of the double angle is important in simplification, integration, and solving equations in trigonometry and calculus.
Given Data / Assumptions:
Concept / Approach:
The standard double angle identity for cosine is:
cos 2A = 2 cos^2 A - 1
From this, we can solve for cos^2 A and then take the square root to express cos A. We keep the factor 1/2 in the denominator, which is crucial for the correct identity.
Step-by-Step Solution:
Start with cos 2A = 2 cos^2 A - 1
Rearrange to isolate cos^2 A:
2 cos^2 A = 1 + cos 2A
So cos^2 A = (1 + cos 2A) / 2
Since cos A = x, we have x^2 = (1 + cos 2A) / 2
Taking square roots, x = √[(1 + cos 2A) / 2], choosing the appropriate sign for the context
Verification / Alternative check:
Take a convenient angle such as A = 60 degrees. Then cos 60 = 1/2. On the right side, cos 2A = cos 120 = -1/2. So (1 + cos 2A)/2 = (1 - 1/2)/2 = (1/2)/2 = 1/4. The square root is 1/2, which matches cos 60. This numerical check supports the derived identity.
Why Other Options Are Wrong:
Option b corresponds to a formula for sin A rather than cos A. Options c and d involve sin 2A, which is related to sin A and cos A but not equal to cos A in this form. Option e omits the division by 2, which changes the value significantly and does not match the standard double angle identity.
Common Pitfalls:
A common mistake is to mix up the formulas for cos 2A and sin 2A or to forget the factor 1/2 when solving for cos^2 A. Another issue is ignoring the sign of the square root. In exam problems like this, the principal root is usually assumed, but in general one must consider the quadrant of angle A.
Final Answer:
The correct double angle expression for x is x = √[(1 + cos 2A)/2].
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