Download Trig Equations 4 - Sum Difference Identities_1

Math 30-1
Trigonometric Equations: Lesson #4
Sum & Difference Identities
Objective: By the end of this lesson, you will be able to:
Question: Does sin      sin   sin  ?
Try it out for   60 and   30 ?
Answer:
The sum and difference identities for sine and cosine are:
e.g. 1) Verify the identity cos     cos  cos   sin  sin  for  
2

and   .
3
6
Math 30-1
Trigonometric Equations: Lesson #4
Tangent also has sum and difference identities, which can be derived from the sine and cosine
sum and difference identities as follows:
Tangent Sum Identity:
sin 
tan  
, so:
cos 
tan     
Substitute in the identities:
Divide all terms by
cos  cos  :
Simplify:
sin 
identity
cos 
to switch back to tangent:
Use the tan  
Similarly, the Tangent Difference Identity is:
e.g. 2) Write the following as a single trigonometric function. Then write the exact value, if
possible.
tan 58  tan 77
a)
b) cos 206 cos133  sin 206 sin 133
1  tan 58 tan 77
Math 30-1
Trigonometric Equations: Lesson #4


e.g. 3) Use sum and difference identities to simplify sin   x  .
2

3
5

3
 B  2 , find the exact value of
and cos B  , where 0  A  and
5
13
2
2
cos A  B .
e.g. 4) If cos A 
Sum and difference identities can also be used to determine exact values of some trig ratios that
are not in the special triangles. To solve these problems, you need to find a combination of the
memorized angles that will give a sum or difference of the angle in the question.
It will always be a combination of angles with reference angles of:
Math 30-1
Trigonometric Equations: Lesson #4
e.g. 5) Determine the exact value of:
a) cos 255
b) tan
Assignment:

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p. 306-307 #1abd, 2abd, 6-10, 11c, 19, 20ab