Correct Answer:
A. 4/5
The correct answer is A: 4/5. To find the cosine of an angle when the sine is known, and the angle is in the first quadrant, we use the fundamental trigonometric identity: sin²θ + cos²θ = 1. Given that sin θ = 3/5, we can substitute this value into the identity.
- First, square sin θ: (3/5)² = 9/25.
- Then, substitute into the identity: 9/25 + cos²θ = 1.
- Subtract 9/25 from both sides: cos²θ = 1 - 9/25 = 25/25 - 9/25 = 16/25.
- Finally, take the square root of both sides: cos θ = √(16/25). This gives two possible values: +4/5 and -4/5.
- Since the problem states that the angle θ is in the first quadrant, both sine and cosine values are positive. Therefore, cos θ = 4/5.
Let's look at the incorrect options:
- Option B: 3/5 is incorrect because this is the value given for sin θ, not cos θ. The two functions are different and generally have different values for the same angle.
- Option C: 5/3 is incorrect for two reasons. Firstly, the value of sine or cosine can never be greater than 1 (or less than -1). 5/3 is approximately 1.67, which is outside the valid range. Secondly, it is the reciprocal of a side ratio, not a correct calculation for cos θ.
- Option D: 1/2 is incorrect because it does not satisfy the Pythagorean identity. If cos θ = 1/2, then (3/5)² + (1/2)² = 9/25 + 1/4 = 36/100 + 25/100 = 61/100, which is not equal to 1.