Download Sum and Difference Formulas for Sine, Cosine and Tangent

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
tan( A + B ) =
tan A + tan B
1 − tan A tan B
tan( A − B ) =
tan A − tan B
1 + tan A tan B
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Example 1: Simplify each:
a. cos( x + 60°) =
π
π


b. sin  x +  − sin  x − 
4
4


2
3π

Example 2: Given that tan( x) = 5 , evaluate tan  x +
4


.

3
Example 3: Simplify each.
a. sin10° cos 55° − sin 55° cos10°
π 
 7π
b. cos   cos 
 12 
 12
c.
tan 40° + tan 5°
1 − tan 40° tan 5°
d.
tan 80° − tan15°
1 + tan 80° tan15°

 π   7π 
 + sin   sin 


 12   12 
4
Example 4: Find the exact value of each. (Hint: use sum/difference formulas)
a. sin 15 0
 7π 
b. cos

 12 
5
 5π 
c. tan 

 12 
6
Example 5: Suppose that sin α =
3
5
π
and cos β =
where 0 < α < β < . Find each of
5
13
2
these:
a. sin(α + β )
b. cos(α − β )
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Example 6: Suppose cos α =
1
7
and tan β = − where π < α , β < 2π . Find
5
6
a. cos(α + β )
b. tan(α + β )
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Section 6.2 – Double and Half Angle Formulas
Now suppose we are interested in finding sin(2 A) . We can use the sum formula for sine
to develop this identity:
sin(2 A) = sin( A + A)
= sin A cos A + sin A cos A
= 2 sin A cos A
Similarly, we can develop a formula for cos(2 A) :
cos(2 A) = cos( A + A)
= cos A cos A − sin A sin A
= cos 2 A − sin 2 A
We can restate this formula in terms of sine only or in terms of cosine only by using the
Pythagorean theorem and making a substitution. So we have:
cos(2 A) = cos 2 − sin 2 A
= 1 − 2 sin 2 A
= 2 cos 2 A − 1
We can also develop a formula for tan(2 A) :
tan(2 A) = tan( A + A)
tan A + tan A
=
1 − tan A tan A
2 tan A
=
1 − tan 2 A
These three formulas are called the double angle formulas for sine, cosine and tangent.
1
Double – Angle Formulas
sin(2 A) = 2 sin A cos A
cos(2 A) = cos 2 A − sin 2 A
tan(2 A) =
(Also: cos(2 A) = 2 cos 2 A − 1 = 1 − 2 sin 2 A )
2 tan A
1 − tan 2 A
Now we’ll look at the types of problems we can solve using these identities.
2
4
π
Example 1: Suppose that cos α = − and < α < π . Find
7
2
a. cos(2α )
b. sin(2α )
c. tan(2α )
3
Example 2: Simplify each:
a. 2 sin 45° cos 45°
b. cos 2
π
9
− sin 2
π
9
4
c.
2 tan15°
1 − tan 2 15°
d. 1 − 2 sin 2 (6 A)
5
Half – Angle Formulas
1 − cos A
 A
sin   = ±
2
2
1 + cos A
 A
cos  = ±
2
2
sin A
1 − cos A
 A
tan   =
=
sin A
 2  1 + cos A
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
A
which the angle
terminates.
2
Now we’ll look at the kinds of problems we can solve using half-angle formulas.
6
Example 3: Use a half-angle formula to find the exact value of each.
a. sin 15°
 5π 
b. cos 

 8 
7
 7π 
c. tan  
 12 
8
Example 4: Answer these questions for cos θ =
4 3π
,
< θ < 2π .
9 2
a. In which quadrant does the terminal side of the angle lie?
b. Complete the following: ___ <
θ
2
< ___
c. In which quadrant does the terminal side of
θ
2
lie?
θ 
d. Determine the sign of sin   .
2
θ 
e. Determine the sign of cos  .
2
θ 
f. Find the exact value of sin   .
2
θ 
g. Find the exact value of cos  .
2
θ 
h. Find the exact value of tan  .
 2
9
Section 6.3 - Solving Trigonometric Equations
Next, we’ll use all of the tools we’ve covered in our study of trigonometry to solve some
equations. An equation that involves a trigonometric function is called a trigonometric
equation. Since trigonometric functions are periodic, there may be infinitely solutions to some
trigonometric equations.
Let’s say we want to solve the equation:
The first angles that come to mind are: x =
sin( x) =
π
6
1
2
and x =
5π
.
6
Remember that the period of the sine function is 2π ; sine function repeats itself after each
rotation.
Therefore, the solutions of the equation are: x =
π
6
+ 2kπ , x =
5π
+ 2kπ , where k is any
6
integer.
Recall: For sine and cosine functions, the period is 2π . For tangent and cotangent functions,
the period is π .
1
Example 1: a) Solve the equation in the interval [0,2π ) :
2 cos x = −1
b) Find all solutions to the equation: 2 cos x = −1
Example 2: a) Solve the equation in the interval [0, π ) :
tan x = −1
b) Find all solutions to the equation: tan x = −1
2
Example 3: Solve the equation in the interval [0, π ) :
2sin(2 x) = 1
Example 4: Solve the equation in the interval [0,2π ) :
csc 2 x = 4
3
Example 5: Find all solutions to the equation: cos(2 x) = 0
4
Example 6: Solve the equation in the interval [0,2π ) :
2 sin 2 x − 5 sin x − 3 = 0
5
Example 7: Solve the equation in the interval [0,2π ) :
cos 2 x − 3sin x − 3 = 0
6
Example 8: Solve the equation in the interval [0,2π ) :
cos(2 x) = 5 sin 2 x − cos 2 x
7
Example 9: Find all solutions to the equation:
sin 2 x cos x = cos x
8
Example 10: Find all solutions:
sec 2 x + 2 tan x = 0
9
Example 11: Solve the equation in the interval [0, 2) :
cot(π x) = −1
Example 12: Find all solutions of the equation in the interval [0, 4π ) :
 x
2 sin   = 1
2
10
Example 13: Find all solutions of the equation in the interval [0,2π ) :
sec( x + 2π ) = 2
11
Example 14: Find all solutions of the equation in the interval [0, π ) :
3π 

2 sin  2 x −
= 2
2 

12