Use the given information to find the exact value of each of the following: sin⁡2θ\sin2\thetacos⁡θ=2425,θ lies in quadrant IV.\cos \theta = \frac{24}{25}, \quad \theta \text{ lies in quadrant IV.}

Ch. 3 - Trigonometric Identities and Equations

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Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

문제 9

3장, 문제 9

Use the given information to find the exact value of each of the following: $sine 2 theta$
$\cos θ = \frac{24}{25},$

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Identify the given information: $\cos \theta = \frac{24}{25}$ and $\theta$ lies in quadrant IV.

Recall that in quadrant IV, cosine is positive and sine is negative. Use the Pythagorean identity to find $\sin \theta$: $\sin^2 \theta + \cos^2 \theta = 1$.

Substitute the known value of $\cos \theta$ into the identity: $\sin^2 \theta + \left(\frac{24}{25}\right)^2 = 1$.

Solve for $\sin \theta$: $\sin^2 \theta = 1 - \left(\frac{24}{25}\right)^2$, then take the square root and choose the negative root because $\theta$ is in quadrant IV.

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

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Trigonometric Ratios and Quadrants

Trigonometric ratios like sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. The quadrant in which the angle lies determines the sign (positive or negative) of these ratios. For example, in quadrant IV, cosine is positive while sine is negative.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This identity allows you to find the sine value when cosine is known, by rearranging to sin θ = ±√(1 - cos²θ). The sign depends on the quadrant of θ.

Double-Angle Formula for Sine

The double-angle formula for sine is sin 2θ = 2 sin θ cos θ. It expresses the sine of twice an angle in terms of the sine and cosine of the original angle. This formula is essential for finding sin 2θ once sin θ and cos θ are known.

Use the given information to find the exact value of each of the following: sin2θ\(\sin\)2\(\theta\)cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)