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Formulas and examples for finding the values of sine, cosine and tangent functions at double and half angles. It also covers solving trigonometric equations.
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Section 6.1 - Sum and Difference Formulas
Note: sin( A + B )≠sin( A )+sin( B )
cos( A + B )≠cos( A )+cos( B )
Sum and Difference Formulas for Sine, Cosine and Tangent
sin( A + B )=sin A cos B +sin B cos A
sin( A − B )=sin A cos B −sin B cos A
cos( A + B )=cos A cos B −sin A sin B
cos( A − B )=cos A cos B +sin A sin B
1 tan tan
tan tan tan( ) −
1 tan tan
tan tan tan( )
Example 1: Simplify each:
a. cos( x + 60 )° =
b. sin sin 4 4
x x
Example 3 : Simplify each.
a. sin10 cos 55° ° − sin 55 cos10° °
b.
cos cos sin sin 12 12 12 12
c. − ° °
1 tan 40 tan 5
tan 40 tan 5
d.
tan 80 tan
1 tan 80 tan
Example 4: Find the exact value of each. (Hint: use sum/difference formulas)
a. 0 sin 15
b. (^)
cos
Example 5: Suppose that 5
these:
Example 6: Suppose
cos 5
tan 6
Double – Angle Formulas
sin( 2 A )= 2 sin A cos A
2 2 cos( 2 )= cos −sin (Also: A A A 2 2 cos( 2 )= 2 cos − 1 = 1 − 2 sin )
2 1 tan
2 tan tan( 2 ) −
Now we’ll look at the types of problems we can solve using these identities.
Example 1: Suppose that
cos 7
. Find
c. (^2)
2 tan
1 tan 15
d. 1 2 sin ( 6 )
2 − A
Half – Angle Formulas
1 cos
2
sin
1 cos
2
cos
sin
1 cos
1 cos
sin
2
tan
Note: In the half-angle formulas the ± symbol is intended to mean either positive or
negative but not both, and the sign before the radical is determined by the quadrant in
which the angle 2
terminates.
Now we’ll look at the kinds of problems we can solve using half-angle formulas.
c. (^)
tan
cos = < <.
a. In which quadrant does the terminal side of the angle lie?
b. Complete the following: ___ 2
c. In which quadrant does the terminal side of 2
lie?
d. Determine the sign of (^)
sin
e. Determine the sign of (^)
cos
f. Find the exact value of (^)
sin
g. Find the exact value of (^)
cos
h. Find the exact value of (^)
tan
2 csc x = 4
