d
A
B
=
(
cos
(
α
−
β
)
−
1
)
2
+
(
sin
(
α
−
β
)
−
0
)
2
=
cos
2
(
α
−
β
)
−
2
cos
(
α
−
β
)
+
1
+
sin
2
(
α
−
β
)
Applying the Pythagorean identity and simplifying we get:
=
(
cos
2
(
α
−
β
)
+
sin
2
(
α
−
β
)
)
−
2
cos
(
α
−
β
)
+
1
=
1
−
2
cos
(
α
−
β
)
+
1
=
2
−
2
cos
(
α
−
β
)
Because the two distances are the same, we set them equal to each other and simplify.
2
−
2
cos
α
cos
β
−
2
sin
α
sin
β
=
2
−
2
cos
(
α
−
β
)
2
−
2
cos
α
cos
β
−
2
sin
α
sin
β
=
2
−
2
cos
(
α
−
β
)
Finally we subtract
2
from both sides and divide both sides by
−2.
cos
α
cos
β
+
sin
α
sin
β
=
cos
(
α
−
β
)
Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.
A General Note
These formulas can be used to calculate the cosine of sums and differences of angles.
cos
(
α
+
β
)
=
cos
α
cos
β
−
sin
α
sin
β
cos
(
α
−
β
)
=
cos
α
cos
β
+
sin
α
sin
β
How To
Given two angles, find the cosine of the difference between the angles.
Write the difference formula for cosine.
Substitute the values of the given angles into the formula.
Simplify.
Using the formula for the cosine of the difference of two angles, find the exact value of
cos
(
5
π
4
−
π
6
)
.
Begin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.
cos
(
α
−
β
)
=
cos
α
cos
β
+
sin
α
sin
β
cos
(
5
π
4
−
π
6
)
=
cos
(
5
π
4
)
cos
(
π
6
)
+
sin
(
5
π
4
)
sin
(
π
6
)
=
(
−
2
2
)
(
3
2
)
−
(
2
2
)
(
1
2
)
=
−
6
4
−
2
4
=
−
6
−
2
4
Keep in mind that we can always check the answer using a graphing calculator in radian mode.
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Find the exact value of
cos
(
75°
)
.
As
75°
=
45°
+
30°
, we can evaluate
cos
(
75°
)
as
cos
(
45°
+
30°
)
.
cos
(
α
+
β
)
=
cos
α
cos
β
−
sin
α
sin
β
cos
(
45°
+
30°
)
=
cos
(
45°
)
cos
(
30°
)
−
sin
(
45°
)
sin
(
30°
)
=
2
2
(
3
2
)
−
2
2
(
1
2
)
=
6
4
−
2
4
=
6
−
2
4
Keep in mind that we can always check the answer using a graphing calculator in degree mode.
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The
sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.
A General Note
These formulas can be used to calculate the sines of sums and differences of angles.
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
sin
(
α
−
β
)
=
sin
α
cos
β
−
cos
α
sin
β
How To
Given two angles, find the sine of the difference between the angles.
Write the difference formula for sine.
Substitute the given angles into the formula.
Simplify.
Using sum and difference identities to evaluate the difference of angles
Use the sum and difference identities to evaluate the difference of the angles and show that part
a equals part
b.
sin
(
45°
−
30°
)
sin
(
135°
−
120°
)
Let’s begin by writing the formula and substitute the given angles.
sin
(
α
−
β
)
=
sin
α
cos
β
−
cos
α
sin
β
sin
(
45°
−
30°
)
=
sin
(
45°
)
cos
(
30°
)
−
cos
(
45°
)
sin
(
30°
)
Next, we need to find the values of the trigonometric expressions.
sin
(
45°
)
=
2
2
,
cos
(
30°
)
=
3
2
,
cos
(
45°
)
=
2
2
,
sin
(
30°
)
=
1
2
Now we can substitute these values into the equation and simplify.
sin
(
45°
−
30°
)
=
2
2
(
3
2
)
−
2
2
(
1
2
)
=
6
−
2
4
Again, we write the formula and substitute the given angles.
sin
(
α
−
β
)
=
sin
α
cos
β
−
cos
α
sin
β
sin
(
135°
−
120°
)
=
sin
(
135°
)
cos
(
120°
)
−
cos
(
135°
)
sin
(
120°
)
Next, we find the values of the trigonometric expressions.
sin
(
135°
)
=
2
2
,
cos
(
120°
)
=
−
1
2
,
cos
(
135°
)
=
2
2
,
sin
(
120°
)
=
3
2
Now we can substitute these values into the equation and simplify.
sin
(
135°
−
120°
)
=
2
2
(
−
1
2
)
−
(
−
2
2
)
(
3
2
)
=
−
2
+
6
4
=
6
−
2
4
sin
(
135°
−
120°
)
=
2
2
(
−
1
2
)
−
(
−
2
2
)
(
3
2
)
=
−
2
+
6
4
=
6
−
2
4
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