Use the given information to find the exact value of each of the following: sin⁡2θ\sin2\theta sin⁡θ=1213,θ lies in quadrant II.\sin \theta = \frac{12}{13}, \quad \theta \text{ lies in quadrant II.}

Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

문제 8a

3장, 문제 8a

Use the given information to find the exact value of each of the following: $sine 2 theta$
$\sin θ = \frac{12}{13},$

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Identify the given information: $\sin \theta = \frac{12}{13}$ and $\theta$ lies in quadrant II.

Recall the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. Use this to find $\cos \theta$.

Calculate $\cos \theta$ by rearranging the identity: $\cos \theta = \pm \sqrt{1 - \sin^2 \theta} = \pm \sqrt{1 - \left(\frac{12}{13}\right)^2}$.

Determine the correct sign of $\cos \theta$ based on the quadrant. Since $\theta$ is in quadrant II, $\cos \theta$ is negative.

Use the double-angle formula for sine: $\sin 2\theta = 2 \sin \theta \cos \theta$. Substitute the known values of $\sin \theta$ and $\cos \theta$ to express $\sin 2\theta$.

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Understanding the Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each determining the sign of sine, cosine, and tangent functions. In quadrant II, sine values are positive while cosine and tangent values are negative. Knowing the quadrant helps determine the correct sign of trigonometric values when solving problems.

Double-Angle Identity for Sine

The double-angle identity for sine states that sin(2θ) = 2 sin(θ) cos(θ). This formula allows you to find the sine of twice an angle using the sine and cosine of the original angle. It is essential for problems requiring exact values of trigonometric functions at multiple angles.

Using the Pythagorean Identity to Find Cosine

Given sin(θ), the cosine can be found using the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Rearranging gives cos(θ) = ±√(1 - sin²(θ)). The sign depends on the quadrant of θ, which is crucial for determining the correct cosine value to use in calculations.

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\) sinθ=1213,θ lies in quadrant II.\(\sin\) \(\theta\) = \(\frac{12}{13}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant II.}\)