Given cos60∘=12, use the trigonometric identitis to find the exact value of each of the following
The Correct Answer and Explanation is:
Correct Answer: 1/2
Explanation:
The problem asks to find the exact value of sin 30° using the given information that cos 60° = 1/2 and by applying trigonometric identities. The key to solving this is to use the cofunction identities, which relate trigonometric functions of complementary angles. Complementary angles are two angles that add up to 90 degrees.
The relevant cofunction identity for this problem relates the sine and cosine functions:
sin(θ) = cos(90° – θ)
This identity states that the sine of an angle θ is equal to the cosine of its complement, 90° – θ. We can use this relationship to find the value of sin 30°.
- Apply the Cofunction Identity: Let’s set the angle θ to be 30°. According to the identity, we have:
sin(30°) = cos(90° – 30°) - Simplify the Expression: Next, we perform the subtraction inside the cosine function:
90° – 30° = 60°
This simplifies our equation to:
sin(30°) = cos(60°) - Use the Given Value: The problem provides the exact value for cos 60°. We are given:
cos 60° = 1/2 - Determine the Final Answer: Since we have established that sin 30° is equal to cos 60°, we can substitute the given value into our equation:
sin 30° = 1/2
Therefore, by using the cofunction identity, we find that the exact value of sin 30° is 1/2. This result is a simplified fraction as required by the problem instructions.
