Question - Trigonometry Question: Finding Cosine Value Given Sine
Solution:
The image shows a trigonometry question which reads:"Question 26You 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)
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