Example Question - exact value
Here are examples of questions we've helped users solve.
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.
Trigonometry Problem: Calculating Tangent and Cosine
The image contains a trigonometry problem which states: "(b) Given that \( \sin\theta = \frac{3}{5} \) and \( 90° \leq \theta \leq 180° \), find the exact value of \( \tan\theta + \cos\theta \)." Since \( \sin\theta = \frac{3}{5} \) and the angle \( \theta \) falls in the second quadrant (because \( 90° \leq \theta \leq 180° \)), we know that the cosine of \( \theta \) will be negative, and the tangent of \( \theta \) will be negative as well because the tangent of an angle is the sine divided by the cosine, and in the second quadrant the cosine is negative. We can find the cosine of \( \theta \) using the Pythagorean identity, where \( \cos^2\theta = 1 - \sin^2\theta \). \( \cos^2\theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \) Since \( \theta \) is in the second quadrant, \( \cos\theta \) will be negative: \( \cos\theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \) Now we can find the tangent of \( \theta \): \( \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4} \) Now, we can find the value of \( \tan\theta + \cos\theta \): \( \tan\theta + \cos\theta = -\frac{3}{4} + (-\frac{4}{5}) = -\frac{3 \cdot 5}{4 \cdot 5} - \frac{4 \cdot 4}{4 \cdot 5} = -\frac{15}{20} - \frac{16}{20} = -\frac{31}{20} \) So, the exact value of \( \tan\theta + \cos\theta \) given \( \sin\theta = \frac{3}{5} \) and \( 90° \leq \theta \leq 180° \) is \( -\frac{31}{20} \).
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