Domanda del libro di testo
In Exercises 7–14, use the given information to find the exact value of each of the following:
b. cos 2θ
15
sin θ = -------- , θ lies in quadrant II.
17
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 7
In Exercises 7–14, use the given information to find the exact value of each of the following: a. sin 2θ 15 sin θ = -------- , θ lies in quadrant II. 17
Guida verificata passo dopo passo1
Recognize that \( \sin \theta = \frac{15}{17} \) and \( \theta \) is in Quadrant II, where sine is positive and cosine is negative.
Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).
Substitute \( \sin \theta = \frac{15}{17} \) into the identity: \( \left(\frac{15}{17}\right)^2 + \cos^2 \theta = 1 \).
Solve for \( \cos^2 \theta \) and then find \( \cos \theta \). Remember, since \( \theta \) is in Quadrant II, \( \cos \theta \) will be negative.
Use the double angle formula for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \) to find \( \sin 2\theta \).
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Sine Function and Its Values
The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. In this problem, sin(θ) is given as 15/17, which indicates that for angle θ in quadrant II, the sine value is positive while the cosine value is negative. Understanding the sine function's behavior in different quadrants is crucial for solving trigonometric problems.
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5:08
Sine, Cosine, & Tangent of 30°, 45°, & 60°
Double Angle Formula for Sine
The double angle formula for sine states that sin(2θ) = 2sin(θ)cos(θ). This formula allows us to find the sine of double an angle using the sine and cosine of the original angle. To apply this formula, we need to calculate cos(θ) using the Pythagorean identity, which relates sine and cosine values.
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05:06Double Angle Identities
Pythagorean Identity
The Pythagorean identity states that sin²(θ) + cos²(θ) = 1. This identity is essential for finding the cosine value when the sine value is known. In this case, since sin(θ) = 15/17, we can use this identity to calculate cos(θ) and subsequently find sin(2θ) using the double angle formula.
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6:25Pythagorean Identities
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