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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 14a
Capitolo 3, Problema 14a

Use the given information to find the exact value of each of the following: sin 2θ
sin θ = ﹣2/3, θ lies in quadrant III.

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Identify the given information: \(\sin \theta = -\frac{2}{3}\) and \(\theta\) lies in quadrant III. Recall that in quadrant III, both sine and cosine are negative.
Use the Pythagorean identity to find \(\cos \theta\): \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\sin \theta = -\frac{2}{3}\) to get \(\left(-\frac{2}{3}\right)^2 + \cos^2 \theta = 1\).
Simplify the equation: \(\frac{4}{9} + \cos^2 \theta = 1\), then solve for \(\cos^2 \theta\) to find \(\cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9}\).
Determine the sign of \(\cos \theta\) in quadrant III. Since cosine is negative in quadrant III, \(\cos \theta = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3}\).
Use the double-angle formula for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\). Substitute the values found: \(\sin 2\theta = 2 \times \left(-\frac{2}{3}\right) \times \left(-\frac{\sqrt{5}}{3}\right)\).

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Double-Angle Identity for Sine

The double-angle identity for sine states that sin(2θ) = 2 sin(θ) cos(θ). This formula allows you to find the sine of twice an angle using the sine and cosine of the original angle, which is essential for solving problems involving sin 2θ.
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Determining the Sign of Trigonometric Functions by Quadrant

The sign of sine, cosine, and tangent depends on the quadrant in which the angle lies. Since θ is in quadrant III, both sine and cosine are negative. This information helps determine the correct signs of trigonometric values when calculating sin 2θ.
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Introduction to Trigonometric Functions

Using the Pythagorean Identity to Find Cosine

Given sin θ, the Pythagorean identity sin²θ + cos²θ = 1 allows you to find cos θ by rearranging to cos θ = ±√(1 - sin²θ). The sign of cos θ is chosen based on the quadrant of θ, which is crucial for accurately computing sin 2θ.
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