Question

Find values of the sine and cosine functions for each angle measure.B, given cos 2B = 1/8 , 540° \u003c 2B \u003c 720°

Accepted Answer

Identify the given information: $$ \cos 2B = \frac{1}{8} $$ and the angle range $$ 540^\circ \u003c 2B \u003c 720^\circ . $$This means $$ 2B  is in $$the third or fourth quadrant since $$ 540^\circ = 360^\circ + 180^\circ $$ and $$ 720^\circ = 2 	imes 360^\circ . $$Determine the quadrant of $$ 2B $$ based on the given range. Since $$ 540^\circ \u003c 2B \u003c 720^\circ $$, $$ 2B $$ 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. Use the double-angle identity for cosine: $$ \cos 2B = 2 \cos^2 B - 1 . $$Substitute $$ \cos 2B = \frac{1}{8}  to $$get the equation $$ \frac{1}{8} = 2 \cos^2 B - 1 . $$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, $$ \cos B = \pm \frac{3}{4} . $$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.

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

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Concetti chiave

Double-Angle Identities

Angle Measurement and Quadrants

Inverse Trigonometric Functions and Solving Equations

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