Use the given information to find the exact value of each of the following: tan⁡2θ\tan2\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 9c

3장, 문제 9c

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

검증된 단계별 안내

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 $\cos \theta = \frac{24}{25}$ to find $\sin \theta$.

Calculate $\sin \theta$ by rearranging the identity: $\sin \theta = -\sqrt{1 - \left(\frac{24}{25}\right)^2}$, taking the negative root because $\sin \theta$ is negative in quadrant IV.

Use the double-angle formula for tangent: $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$. To apply this, first find $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Substitute $\tan \theta$ into the double-angle formula to express $\tan 2\theta$ in terms of known values, then simplify the expression to find the exact value.

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Trigonometric Ratios and Quadrants

Trigonometric ratios like sine, cosine, and tangent relate the angles of a right triangle to side lengths. Knowing the quadrant of the angle is crucial because it determines the sign (positive or negative) of these ratios. In quadrant IV, cosine is positive while sine and tangent are negative.

Double-Angle Identity for Tangent

The double-angle identity for tangent states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This formula allows you to find the tangent of twice an angle using the tangent of the original angle, which is essential for solving the problem once tan θ is known.

Finding tan θ from cos θ

Given cos θ, you can find sin θ using the Pythagorean identity sin² θ + cos² θ = 1. Then, tan θ is calculated as sin θ divided by cos θ. The quadrant information helps determine the correct sign of sin θ and tan θ.

Use the given information to find the exact value of each of the following: tan2θ\(\tan\)2\(\theta\)cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)