Find values of the sine and cosine functions for each angle measure.
B, given cos 2B = 1/8 , 540° < 2B < 720°

Verified step by step guidance

1. Identify the given information: \( \cos 2B = \frac{1}{8} \) and the angle range \( 540^\circ < 2B < 720^\circ \). This means \( 2B \) is in the third or fourth quadrant since \( 540^\circ = 360^\circ + 180^\circ \) and \( 720^\circ = 2 \times 360^\circ \).
2. Determine the quadrant of \( 2B \) based on the given range. Since \( 540^\circ < 2B < 720^\circ \), \( 2B \) lies between \( 180^\circ \) and \( 360^\circ \) after subtracting \( 360^\circ \) (because cosine is periodic with period \( 360^\circ \)). This places \( 2B \) effectively in the third or fourth quadrant of the unit circle.
3. Use the double-angle identity for cosine: \( \cos 2B = 2 \cos^2 B - 1 \). Substitute \( \cos 2B = \frac{1}{8} \) to get the equation \( \frac{1}{8} = 2 \cos^2 B - 1 \).
4. Solve for \( \cos B \) by isolating \( \cos^2 B \): \[ 2 \cos^2 B = 1 + \frac{1}{8} = \frac{9}{8} \implies \cos^2 B = \frac{9}{16} \]. Then, \( \cos B = \pm \frac{3}{4} \).
5. Determine the correct sign of \( \cos B \) by considering the quadrant of \( B \). Since \( 2B \) is between \( 540^\circ \) and \( 720^\circ \), \( B \) is between \( 270^\circ \) and \( 360^\circ \), which is the fourth quadrant where cosine is positive and sine is negative. Use the Pythagorean identity \( \sin^2 B = 1 - \cos^2 B \) to find \( \sin B \), taking the negative root for sine.

Key Concepts

Double-Angle Identities

Double-angle identities relate trigonometric functions of an angle to functions of twice that angle. For cosine, cos(2B) can be expressed as 2cos²(B) - 1 or 1 - 2sin²(B). These identities help find sine and cosine values of B when cos(2B) is known.

Angle Measurement and Quadrants

Understanding the angle's quadrant is essential because sine and cosine signs depend on the quadrant. Since 540° < 2B < 720°, 2B lies in the third or fourth revolution, affecting the possible values and signs of sine and cosine for B.

Inverse Trigonometric Functions and Solving Equations

To find angle B from cos(2B) = 1/8, inverse cosine is used. Solving for B requires manipulating the equation and considering the periodicity of trigonometric functions to find all valid solutions within the given interval.