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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 9b
Capitolo 3, Problema 9b

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.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)

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1
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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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.
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Quadratic Formula

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.
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Double Angle Identities

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.
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Pythagorean Identities
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cosθ=2425,θ lies in quadrant IV.\(\cos\) \(\theta\) = \(\frac{24}{25}\), \(\quad\) \(\theta\) \(\text{ lies in quadrant IV.}\)

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