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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 9c
Capitolo 3, Problema 9c

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

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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 \(\cos \theta = \frac{24}{25}\) 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.
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Quadratic Formula

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.
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Double Angle Identities

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.
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Finding Components from Direction and Magnitude
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