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Trigonometry Part III
Half Angles
Formulas for Half Angles
Half Angles
1
sin
2
1
cos
2
1
cos
2
1
cos
2
1
tan
2
1
1
cos
cos
1
Trigonometry Part III
*Note: The choice for the sign
on the outside of the radical
depends upon the location of
. The choice for the sign on
the inside of the radical for
cos depends upon the
location of angle .
Example 1: If is a positive acute angle and
cos
, then evaluate sin
.
Step 1: Use the correct half angle formula and determine if any
additional values are necessary.
1
sin
2
1
cos
2
*We have the
value we need!
Step 2: Plug-in appropriate values into the formula and simplify.
1
2
Half Angles
7
25∙ 25
25
25 7
50
18
50
9
25
3
5
2
Trigonometry Part III
Example 2: If is located in Quadrant III and
tan
, then evaluate cos
.
cos
1
2
1
2
1
17
8
B
15
15
17 ∙ 17
17
17
Quadrant III
15
34
1
1
17
2
34
cos
2
17
17
17
Example 3: Use sin
value of sin 60° .
120° to find the exact
Step 1: Determine which formula to use and any
other necessary trigonometric values.
1
sin
2
1
cos 120°
cos 60°
cos
2
1
2
Step 2: Plug the values into the formula and simplify.
1
2
Half Angles
1
2
1
2
1
2
2
∙
2
2
1
4
3
2
3
Trigonometry Part III
Example 4: Use a half angle formula to
determine the exact value of cos 22.5° .
cos
2
cos
2
1
cos 45°
2
cos 22.5°
1
1
1
2
2
2 ∙2
2
1
cos 45°
2
2
2
2
4
2
2
Example 5: Use a half angle formula to
determine the exact value of sin 105° .
sin
sin 105°
1
Half Angles
1
2
sin
cos 30°
2
1
cos
2
1
210°
2
1
2
1
3
2 ∙2
2
cos 210°
2
2
3
4
2
3
2
4