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Trigonometric Identities and Equations
Sum and Difference Identities
It is relatively simple to manipulate trig functions for given angles. But what if we
wanted to find the value of a trig function for an angle that we do not know? Luckily we
have several identities that allow us to do just that.
Sum and Difference Identities for the Cosine Function:
If α and β represent the measures of two angles, then the following
identities hold for all values of α and β.
cos(α + β ) = cos α cos β − sin α sin β
cos(α − β ) = cos α cos β + sin α sin β
In other words we can find the cosine of the sum of two angles by subtracting the
product of the sines of the two angles from the product of the cosines of the
angles. Also we can find the cosine of the difference between two angles by
adding the product of the sines of the two angles to the product of the cosines of
the angles.
Example 1: Find cos15° from values of functions of 30° and 45 °
15 ° is 45° - 30° thus cos15° = cos (45° - 30°)
cos (45° - 30°) = cos 45°cos30° + sin45°sin30° =
2
3
2 1
6
2
•
+
• =
+
≈ .9659
2
2
2 2
4
4
cos15° ≈ .9659
There are also sum and difference identities for the sine and tangent functions.
Sum and Difference Identities for the Sine Function:
If α and β represent the measures of two angles, then the following
identities hold for all values of α and β.
sin(α + β ) = sin α cos β + cos α sin β
sin(α − β ) = sin α cos β − cos α sin β
Example 2:
Find the value of sin75° from values of functions of 30° and 45°
75° = 30° + 45° so sin75° = sin (30° + 45°)
sin(30° + 45°) = sin 30° cos 45° + cos 30° sin 45°
1
2
3
2
2
6
•
+
•
=
+
≈ .9659
2 2
2
2
4
4
Sum and Difference Identities for the Tangent Function:
If α and β represent the measures of two angles, then the following
identities hold for all values of α and β.
Example 3:
tan(α + β ) =
tan α + tan β
1 − tan α tan β
tan(α − β ) =
tan α − tan β
1 + tan α tan β
Find tan15° from values of functions of 45° and 30°.
tan15° = tan(45° − 30°) =
tan 45° − tan 30°
=
1 + tan 45° tan 30°
1
3
3
1−
−
3 = 3 3 = 3− 3 • 3− 3 =
⎛ 1 ⎞ 3
3 3+ 3 3− 3
1 + 1⎜
+
⎟
⎝ 3⎠ 3 3
9 − 6 3 + 3 12 − 6 3
=
= 2 − 3 ≈ .2679
9−3
6