Trigonometry Question: Finding Cosine Value Given Sine
The image shows a trigonometry question which reads:
"Question 26
You are told that sin θ = \(\frac{7}{\sqrt{389}}\).
a) If θ is in the first quadrant, then the exact value of cos θ is __________.
Note: In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example, if your answer is \(\frac{\sqrt{17}}{17}\), then enter sqrt(5)/sqrt(17)"
To solve for cos θ when given sin θ, we can use the Pythagorean identity, which states that sin² θ + cos² θ = 1.
Since we're given sin θ = \(\frac{7}{\sqrt{389}}\), we can square this to find sin² θ:
(sin θ)² = \(\left(\frac{7}{\sqrt{389}}\right)^2 = \frac{49}{389}\)
Next we'll use the identity to solve for cos² θ:
cos² θ = 1 - sin² θ = 1 - \(\frac{49}{389}\)
To find cos² θ, we need to subtract \(\frac{49}{389}\) from 1. Since 1 can be written as \(\frac{389}{389}\), we get:
cos² θ = \(\frac{389}{389} - \frac{49}{389} = \frac{389 - 49}{389} = \frac{340}{389}\)
So, cos² θ = \(\frac{340}{389}\).
We are now looking for the positive square root since θ is in the first quadrant where cosine values are positive:
cos θ = \(\sqrt{\frac{340}{389}}\)
The square root of a fraction is the square root of the numerator over the square root of the denominator:
cos θ = \(\frac{\sqrt{340}}{\sqrt{389}}\)
In the most simplified fractional form, this is the exact value for cos θ. However, note that \(\sqrt{340}\) can be further simplified since 340 has a square factor, which is 4 (2²).
\(\sqrt{340} = \sqrt{4 \cdot 85} = 2\sqrt{85}\)
Therefore, the final simplified answer for cosθ is:
cos θ = \(\frac{2\sqrt{85}}{\sqrt{389}}\)
And in the answer format requested in the question:
2sqrt(85)/sqrt(389)
