Which value is equivalent to sin 42°?
A cos 3º
B cos 45°
C cos 90°
D cos 48°
The correct answer and explanation is :
The correct answer is:
D. cos 48°
Explanation:
To find a value equivalent to sin 42°, we can use a fundamental identity in trigonometry known as the co-function identity:
$$
\sin(\theta) = \cos(90^\circ – \theta)
$$
This identity tells us that the sine of an angle is equal to the cosine of its complement. Two angles are complementary if they add up to 90 degrees.
So let’s apply the identity to sin 42°:
$$
\sin(42^\circ) = \cos(90^\circ – 42^\circ) = \cos(48^\circ)
$$
This means sin 42° is exactly equal to cos 48°.
Now let’s examine the other answer choices:
- A. cos 3°
→ Not correct. $\sin(42^\circ) \neq \cos(3^\circ)$ because 42 + 3 = 45, not 90. They’re not complementary angles. - B. cos 45°
→ Not correct. $\sin(42^\circ) \neq \cos(45^\circ)$. These angles are not complementary. - C. cos 90°
→ Not correct. $\cos(90^\circ) = 0$, and $\sin(42^\circ) \approx 0.6691$, so these are clearly not equal. - D. cos 48°
→ Correct. $\sin(42^\circ) = \cos(90^\circ – 42^\circ) = \cos(48^\circ)$
This identity is particularly useful when simplifying expressions or solving trigonometric equations. It highlights the symmetry in the unit circle and explains how sine and cosine relate to each other. Both functions describe the ratios of sides in a right triangle, and since the angles in a triangle must sum to 180°, the acute angles in a right triangle (the ones besides the right angle) must sum to 90°. This is the basis for the co-function identity.