Find the value of a in the equation: cos (a + 14)° = sin (4a + 6)°

Analyze the question: We have cos of one angle equal to sin of another. We need the value of a.

Concept: The complementary angle identity says cos θ = sin(90° − θ). So if cos(a + 14)° = sin(4a + 6)°, then the two angles must add up to 90° (for acute angles).

Solution:

cos(a + 14)° = sin(90 − (a + 14))° = sin(76 − a)°

Now set the angles equal:

76 − a = 4a + 6 76 − 6 = 4a + a 70 = 5a a = 14

Equivalently: (a + 14) + (4a + 6) = 90, so 5a + 20 = 90, so a = 14.

Check:

• cos(14 + 14)° = cos 28°
• sin(4(14) + 6)° = sin 62°
• sin 62° = cos(90 − 62)° = cos 28° ✓

Checking the other options:

• a = 15: cos 29° vs sin 66° = cos 24°, not equal.
• a = 17: cos 31° vs sin 74° = cos 16°, not equal.
• a = 21: cos 35° vs sin 90° = 1, not equal.

Answer: a = 14, which is option C.

Tip: Whenever you see sin and cos equated, think "complementary angles": add the two angles and set the sum equal to 90°. This turns the problem into a simple linear equation.