If sin(3x − 20°) = cos(3y + 20°) for angles x and y measured in degrees, what is the value of the sum (x + y) in degrees, assuming principal values within the standard acute angle range?

Introduction / Context:This question examines the relationship between sine and cosine through the complementary angle identity. It also tests the ability to equate angles inside trigonometric functions to find a relationship between x and y. We focus on principal values in a typical aptitude exam setting, avoiding complications from periodicity unless explicitly required.

Given Data / Assumptions:

• The equation is sin(3x − 20°) = cos(3y + 20°).
• Angles are in degrees.
• x and y are chosen from a standard principal range so that a simple relationship can be found.
• We must determine the value of x + y in degrees.

Concept / Approach:Use the complementary angle identity cos θ = sin(90° − θ). This allows us to rewrite cos(3y + 20°) as a sine function of a complementary angle. Once both sides are written as sine, we equate the arguments of the sine functions in the simplest principal case. This leads directly to a linear equation in x and y that gives x + y.

Step-by-Step Solution:Start with sin(3x − 20°) = cos(3y + 20°).Use the identity cos θ = sin(90° − θ). Therefore cos(3y + 20°) = sin(90° − (3y + 20°)) = sin(70° − 3y).Thus the equation becomes sin(3x − 20°) = sin(70° − 3y).For principal values in the range usually used in aptitude exams, we equate the arguments directly: 3x − 20° = 70° − 3y.Rearrange: 3x + 3y = 70° + 20° = 90°, so 3(x + y) = 90°.Therefore x + y = 90° / 3 = 30°.

Verification / Alternative check:We could consider the more general sine equality sin α = sin β, which has two primary cases: α = β + 360k° or α = 180° − β + 360k°. The principal simple case α = β gives 3x − 20° = 70° − 3y and leads to x + y = 30°. The alternative case would give a more complicated relationship and is usually not intended in standard multiple choice aptitude questions unless specifically mentioned. Checking with sample values that satisfy x + y = 30° confirms that the equation holds for suitable choices of x and y.

Why Other Options Are Wrong:

• 90°, 60°, and 120° would come from different angle relations not supported by the principal simple identity used.
• 45° does not satisfy the linear equation 3(x + y) = 90° and is not consistent with the complementary relationship.
• Only 30° emerges naturally from the simplest and most standard manipulation.

Common Pitfalls:

• Forgetting to convert cosine to sine using the complementary identity, and instead trying to manipulate both sides separately.
• Misapplying the general sine equality and creating unnecessary complexity.
• Arithmetic mistakes when rearranging 3x − 20° = 70° − 3y to obtain x + y.

Final Answer:30°