In Exercises 7–14, use the given information to find the exact value of each of the following:a. sin 2θ12sin θ = -------- , θ lies in quadrant II.13

Verified step by step guidance

1. Start by recalling the double angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \).
2. Since \( \sin \theta = \frac{12}{13} \) and \( \theta \) is in quadrant II, use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).
3. Substitute \( \sin \theta = \frac{12}{13} \) into the Pythagorean identity: \( \left(\frac{12}{13}\right)^2 + \cos^2 \theta = 1 \).
4. Solve for \( \cos^2 \theta \) and then find \( \cos \theta \). Remember that in quadrant II, \( \cos \theta \) is negative.
5. Substitute the values of \( \sin \theta \) and \( \cos \theta \) into the double angle identity to find \( \sin 2\theta \).

Key Concepts

Sine Function

The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ to the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the context of the unit circle, it represents the y-coordinate of a point on the circle corresponding to the angle θ. Understanding the sine function is crucial for solving problems involving angles and their trigonometric values.

Double Angle Formula

The double angle formula for sine states that sin(2θ) = 2sin(θ)cos(θ). This formula allows us to find the sine of double an angle using the sine and cosine of the original angle. It is particularly useful in problems where the angle is doubled, and knowing how to apply this formula is essential for calculating exact values in trigonometric exercises.

Quadrants and Angle Signs

The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions. In quadrant II, sine values are positive while cosine values are negative. Recognizing the quadrant in which an angle lies is vital for determining the correct signs of sine and cosine, which directly impacts the calculation of trigonometric values like sin(2θ).