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Trigonometry
Question

$$5.4$$ 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. $$\frac { 25 \pi } { 12 } = \frac { 11 \pi } { 6 } + \frac { \pi } { 4 }$$

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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