If the angles are measured in degrees, what is the value of sin(630° + A) + cos A expressed in simplest form?

Introduction / Context:
This trigonometry question tests your understanding of angle reduction and periodicity of sine and cosine functions when angles are measured in degrees. By rewriting a large angle like 630 degrees in terms of a familiar reference angle, you can simplify the expression significantly and recognise cancellation between terms.

Given Data / Assumptions:

• Expression: sin(630° + A) + cos A
• Angles are in degrees.
• A is any real angle for which the functions are defined.

Concept / Approach:
We use the periodicity of sine and cosine and the special angle relationships:
sin(θ + 360°) = sin θ
sin(270° + A) = -cos A
The strategy is to reduce 630° + A into an equivalent angle between 0° and 360°, then use known identities. Once we express sin(630° + A) in terms of cos A, we can add cos A and see whether the terms cancel out or combine to a simpler expression.

Step-by-Step Solution:
Step 1: Simplify the angle 630° + A by subtracting 360° since sine is periodic with period 360°. Step 2: 630° + A - 360° = 270° + A, so sin(630° + A) = sin(270° + A). Step 3: Use the identity sin(270° + A) = -cos A. Step 4: Substitute this into the expression: sin(630° + A) + cos A = -cos A + cos A. Step 5: The two terms cancel each other exactly, giving a result of 0.

Verification / Alternative check:
To verify, pick a sample value for A, for example A = 30°. Compute sin(630° + 30°) = sin 660°, which is equal to sin 300° = -sqrt(3)/2. Now compute cos 30° = sqrt(3)/2. Adding them gives -sqrt(3)/2 + sqrt(3)/2 = 0, confirming that the expression is 0 for this value of A and therefore for all valid values of A.

Why Other Options Are Wrong:
sqrt(3)/2 and 1/2 are specific trigonometric values for certain angles, but they do not involve cancellation of equal and opposite terms. 2/sqrt(3) is not consistent with the behaviour of sine and cosine in this situation. The value 1 would require sin(630° + A) to be equal to 1 - cos A, which does not follow from any standard identity. Only 0 correctly reflects the exact cancellation that occurs after simplification.

Common Pitfalls:
A common mistake is to treat 630° as 2 × 315° and try to use less familiar identities instead of simply subtracting 360°. Another error is misremembering sin(270° + A) as cos A instead of -cos A. Keeping track of the quadrant and the sign of the trigonometric functions is essential when dealing with angles greater than 360°.

Final Answer:
The value of sin(630° + A) + cos A is 0.