In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

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Identify the given information: $\sin \theta = \frac{15}{17}$ and $\theta$ lies in quadrant II.

Recall that in quadrant II, sine is positive and cosine is negative. Use the Pythagorean identity to find $\cos \theta$: $\cos \theta = -\sqrt{1 - \sin^2 \theta}$.

Calculate $\cos \theta$ by substituting $\sin \theta = \frac{15}{17}$ into the identity: $\cos \theta = -\sqrt{1 - \left(\frac{15}{17}\right)^2}$.

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 triangle to the ratios of its sides. Knowing the quadrant of the angle is crucial because it determines the sign (positive or negative) of these ratios. For example, in quadrant II, sine is positive while cosine 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 can be derived from sine and cosine values.

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Double Angle Identities

Finding Cosine and Tangent from Sine

Given sin θ and the quadrant, you can find cos θ using the Pythagorean identity cos² θ = 1 - sin² θ, adjusting the sign based on the quadrant. Then, tan θ is found by dividing sin θ by cos θ. These values are essential to apply the double-angle formula for tangent.

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