Download 15 More Trigonometric Identities

15
More Trigonometric Identities
Concepts:
• The Addition and Subtraction Identities for Sine and Cosine
• The Cofunction Identities for Sine and Cosine
• Identities That You Should Learn
15.1
Addition and Subtraction Identities for Sine and Cosine
Theorem 15.1 (Addition and Subtraction Identities for Sine and Cosine)
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
sin(x − y) = sin(x) cos(y) − cos(x) sin(y)
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
cos(x − y) = cos(x) cos(y) + sin(x) sin(y)
Your textbook contains a proof of subtraction identity for cosine in section 7.2. The proof is
very geometric and uses the distance formula (a.k.a. the Pythagorean Theorem in disguise).
We will not prove the subtraction identity for cosine during lecture, but you may add it
to the list of rules that you can use when proving identities. We will use it to prove the
remaining Addition and Subtraction Identities for Sine and Cosine.
Before proving these identities, we will see how we can use these identities to find exact
values of the trigonometric functions for some angles that are not associated with one of the
two special triangles we have studied.
Example 15.2
Evaluate the six trigonometric functions at θ =
1
5π
.
12
Example 15.3
Evaluate the six trigonometric functions at θ =
13π
.
12
Example 15.4
Use the subtraction identity for cosine to prove that
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
.
Example 15.5
Use the addition identity for cosine to prove that cos(2x) = 1 − 2 sin2 (x).
2
Example 15.6
Prove that sin2 (x) =
15.2
1 − cos(2x)
.
2
Cofunction Identities for Sine and Cosine
Example 15.7
Prove that sin(x) = cos
π
2
− x . (This is known as a cofunction identity.)
Example 15.8
Prove that sin(x + y) = sin(x) cos(y) + cos(x) sin(y).
Example 15.9
In an earlier lecture, we made a conjecture about an identity of the form
sin(2x) =
sin(x) cos(x).
Prove that conjecture.
3
Example 15.10
π
Prove that cos(x) = sin
−x .
2
Theorem 15.11 (Cofunction Identities)
• sin(x) = cos( π2 − x)
• cos(x) = sin( π2 − x)
Example 15.12
Let α be an angle in quadrant III and β be an angle in quadrant IV. If sin(α) = − 52 and
cos(β) = 13 , find the exact values for sin(α + β) and tan(α + β). In which quadrant is the
terminal ray of the angle α + β found? (Assume all angles are in standard position.)
4
15.3
Other Identities
There are a slew of other trigonometric identities. The following are regularly helpful in
Calculus. I expect you to know them.
• tan(x) =
sin(x)
cos(x)
• csc(x) =
1
sin(x)
• sec(x) =
1
cos(x)
• cot(x) =
cos(x)
sin(x)
• sin2 (x) + cos2 (x) = 1.
• tan2 (x) + 1 = sec2 (x).
• 1 + cot2 (x) = csc2 (x).
• sin(−x) = − sin(x).
• cos(−x) = cos(x).
• tan(−x) = − tan(x).
• csc(−x) = − csc(x).
• sec(−x) = sec(x).
• cot(−x) = − cot(x).
• sin(x ± 2πn) = sin(x). (For any integer n)
• cos(x ± 2πn) = cos(x). (For any integer n)
• tan(x ± πn) = tan(x). (For any integer n)
• csc(x ± 2πn) = csc(x). (For any integer n)
• sec(x ± 2πn) = sec(x). (For any integer n)
• cot(x ± πn) = cot(x). (For any integer n)
5
• sin(2x) = 2 sin(x) cos(x)
• cos(2x) = cos2 (x) − sin2 (x).
• cos(2x) = 2 cos2 (x) − 1.
• cos(2x) = 1 − 2 sin2 (x).
• sin2 (x) =
1 − cos(2x)
2
• cos2 (x) =
1 + cos(2x)
2
Several of the identities in this list follow quickly from other identities in the list. For example,
you only need to memorize one of the Pythagorean Identities. The other two Pythagorean
Identities are easy to prove.
There are many other trigonometric identities. You can find several in section 7.3 of your
textbook. You will not be asked to memorize trigonometric identities that are not in the
list above. They will be given to you if you need them. You should be familiar enough with
them, so that you know that they exist and can be found in a table of trigonometric formulas
if you need them. Some of the homework problems may require that you use a formula that
is not in the list. You do need to study them well enough that you know when you need to
look for one of them.
Example 15.13
Express sin2 (x) cos2 (x) in terms of constants and first powers of the cosine function. (NOTE:
This technique will be used in Calculus II.)
6
Example 15.14
Express sin4 (x) in terms of constants and first powers of the cosine function. (NOTE: This
technique will be used in Calculus II.)
7