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Trigonometry
Question

\(5.4\) Evaluating Trigonometric Functions In Exer...

\(5.4\) Evaluating Trigonometric Functions In Exercises \(47 - 50\) , find the exact values of the sine, cosine, and tangent of the angle. 47. \(75 ^ { \circ } = 120 ^ { \circ } - 45 ^ { \circ }\) 

49. \(\frac { 25 \pi } { 12 } = \frac { 11 \pi } { 6 } + \frac { \pi } { 4 }\) 

Answer

47

\(\sin ( 75 ^ { \circ } ) = \frac { \sqrt { 2 + \sqrt { 3 } } } { 2 } ( \) Decimal: \( 0.96592 \ldots ) \) 

 

 

 

Steps 

\( \sin ( 75 ^ { \circ } ) \) 

Use the following identity: \( \sin ( x ) = \cos ( 90 ^ { \circ } - x ) \) 

\( \sin ( 75 ^ { \circ } ) = \cos ( 90 ^ { \circ } - 75 ^ { \circ } ) \) 

\( = \cos ( 90 ^ { \circ } - 75 ^ { \circ } ) \) 

Simplify 

\( = \cos ( 15 ^ { \circ } ) \) 

\( \cos ( 15 ^ { \circ } ) = \frac { \sqrt { 2 + \sqrt { 3 } } } { 2 } \) 

\( = \frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }\) 

 

\(\cos ( 75 ^ { \circ } ) = \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 } ( \) Decimal: \( 0.25881 \ldots ) \) 

Steps 

\( \cos ( 75 ^ { \circ } ) \) 

Use the following identity: \( \cos ( x ) = \sin ( 90 ^ { \circ } - x ) \) 

\( \cos ( 75 ^ { \circ } ) = \sin ( 90 ^ { \circ } - 75 ^ { \circ } ) \) 

\( = \sin ( 90 ^ { \circ } - 75 ^ { \circ } ) \) 

Simplify 

\( = \sin ( 15 ^ { \circ } ) \) 

\( \sin ( 15 ^ { \circ } ) = \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 } \) 

\( = \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }\) 

 

Using the summation identity: \( \tan ( x + y ) = \frac { \tan ( x ) + \tan ( y ) } { 1 - \tan ( x ) \tan ( y ) } \) 

\( = \frac { \tan ( 30 ^ { \circ } ) + \tan ( 45 ^ { \circ } ) } { 1 - \tan ( 30 ^ { \circ } ) \tan ( 45 ^ { \circ } ) } \) 

Use the following trivial identity: \( \tan ( 30 ^ { \circ } ) = \frac { \sqrt { 3 } } { 3 } \) 

Use the following trivial identity: \( \tan ( 45 ^ { \circ } ) = 1 \) 

\( = \frac { \frac { \sqrt { 3 } } { 3 } + 1 } { 1 - \frac { \sqrt { 3 } } { 3 } \cdot 1 } \) 

\( \frac { \sqrt { 3 } } { 3 } + 1 \) 

\( 1 - \frac { \sqrt { 3 } } { 3 } \cdot 1 \) 

\( = 2 + \sqrt { 3 }\) 

Solution
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