Use the given information to find the exact value of each of the following: cos⁡2θ\cos2\theta cos⁡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 9b

3장, 문제 9b

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

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

Recall the double-angle identity for cosine: $\cos 2\theta = 2\cos^{2} \theta - 1$ or equivalently $\cos 2\theta = 1 - 2\sin^{2} \theta$.

Since $\cos \theta$ is given, calculate $\sin \theta$ using the Pythagorean identity $\sin^{2} \theta + \cos^{2} \theta = 1$. Substitute $\cos \theta = \frac{24}{25}$ to find $\sin \theta$.

Determine the sign of $\sin \theta$ in quadrant IV. Remember that in quadrant IV, sine is negative, so take the negative root for $\sin \theta$.

Use the double-angle formula $\cos 2\theta = 2\cos^{2} \theta - 1$ and substitute the known value of $\cos \theta$ to find $\cos 2\theta$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cosine of an Angle in a Specific Quadrant

The cosine function measures the horizontal coordinate on the unit circle. Knowing the quadrant of the angle helps determine the sign of cosine; in quadrant IV, cosine values are positive, which confirms the given positive value of cos θ = 24/25.

Double-Angle Identity for Cosine

The double-angle formula for cosine states that cos 2θ = 2 cos² θ - 1 or cos 2θ = 1 - 2 sin² θ. This identity allows us to find the exact value of cos 2θ using the known value of cos θ or sin θ.

Using Pythagorean Identity to Find sin θ

The Pythagorean identity sin² θ + cos² θ = 1 helps find sin θ when cos θ is known. Since cos θ = 24/25, sin θ can be calculated as ±√(1 - (24/25)²). The sign of sin θ depends on the quadrant; in quadrant IV, sin θ is negative.

Use the given information to find the exact value of each of the following: cos2θ\(\cos\)2\(\theta\) cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)