SUM AND DIFFERENCE IDENTITIES
DOUBLE ANGLE AND HALF ANGLE IDENTITIES
Example 1:
Show that can be changed to sin.
Let's use the Difference Identity for Cosine .
Here, u and v .
Your calculator must be in radian mode!
We showed that equals sin.
Example 2:
Show that can be changed to .
Let's use the Sum Identity for Sine .
Here, u and v .
We showed that equals.
Example 3:
Simplify using the Sum and Difference Identities of Cosine
and .
Evaluate trigonometric ratios as necessary.
Here, u and v x.
We showed that can be simplified to .
Example 4:
Simplify using the Double Angle Identity .
Let's factor out a 3 to get .
We showed that can be written as 3 sin2x.
Example 5:
Simplify using the Double Angle Identity .
Let's factor out a 4 to get
We showed that can be written as 4 cos2x.
Example 6:
Simplify using the Double Angle Identity .
Here, u x.
Let's multiply the parentheses using FOIL to get
We showed that can be written as cos 2x.
Example 7:
Simplify using the Half Angle Identity .
We can see that u 6x, therefore 3x.
Using the pattern of the Half Angle Identity, we can write as sin 3x.
Example 8:
Simplify using the Half Angle Identity.
We can see that u 4x, therefore 2x .
Using the pattern of the Half Angle Identity, we can write as cos 2x.
Example 9:
Rewrite in terms of a Double Angle Identity and a Half Angle Identity:
Using :
We can see that 2u 6x, therefore u 3x.
Using the pattern of the Double Angle Identity, we can write as 2 sin 3x cos 3x.
Using :
We can see that 6x, therefore u 12x.
Using the pattern of the Half Angle Identity, we can write as .