Question

Find values of the sine and cosine functions for each angle measure.2y, given sec y = -5/3, sin y \u003e 0

Accepted Answer

Identify the given information: $$ \sec y = -\frac{5}{3} $$ and $$ \sin y \u003e 0 . $$Recall that $$ \sec y = \frac{1}{\cos y} $$, so first find $$ \cos y  by $$taking the reciprocal of $$ \sec y . $$Determine the quadrant of angle $$ y $$ using the signs of $$ \cos y $$ and $$ \sin y . $$Since $$ \sec y = -\frac{5}{3} $$, $$ \cos y  is $$negative, and $$ \sin y \u003e 0 $$ means sine is positive. Use this to identify the correct quadrant. Use the Pythagorean identity $$ \sin^2 y + \cos^2 y = 1  to $$find $$ \sin y . $$Substitute the value of $$ \cos y $$ found in step 1 into the identity and solve for $$ \sin y $$, choosing the positive root because $$ \sin y \u003e 0 . $$Find the values of $$ \sin 2y $$ and $$ \cos 2y $$ using the double-angle formulas: $$ \sin 2y = 2 \sin y \cos y $$ and $$ \cos 2y = \cos^2 y - \sin^2 y . $$Substitute the values of $$ \sin y $$ and $$ \cos y $$ obtained earlier. Express the final answers for $$ \sin 2y $$ and $$ \cos 2y  in $$simplified form, leaving the expressions in terms of fractions or radicals without calculating decimal approximations.

Find values of the sine and cosine functions for each angle measure.
2y, given sec y = -5/3, sin y > 0

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Key Concepts

Reciprocal Trigonometric Functions

Sign of Trigonometric Functions in Quadrants

Double-Angle Formulas

Find values of the sine and cosine functions for each angle measure.2y, given sec y = -5/3, sin y > 0