Trigonometry Problem Solution: Finding the Exact Value of sin(θ)
The image contains a trigonometry problem. It states:
"Question 27
You are told that cos(θ) = 8/17.
a) If θ is in the first quadrant, then the exact value of sin(θ) is ______.
Note: In this question we require you to input your answer without decimals and without entering the words sin, cos, tan or cot. For example, if your answer is 3/17, then enter sqrt(17^2-3^2)/17)"
To solve this, we will use the Pythagorean identity which relates the sine and cosine of an angle: sin^2(θ) + cos^2(θ) = 1.
We are given that cos(θ) = 8/17. Squaring both sides we get:
cos^2(θ) = (8/17)^2 = 64/289.
Now we can find sin^2(θ):
sin^2(θ) = 1 - cos^2(θ) = 1 - 64/289 = 289/289 - 64/289 = 225/289.
Since we are in the first quadrant, sin(θ) will be positive, so:
sin(θ) = √(225/289) = 15/17.
Therefore, the exact value of sin(θ) is 15/17.
