Transcript Sum and Difference Formulas

5.4
Sum & Difference
Formulas
HWQ
• Solve the equation
2 sin x  1  0
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
2
Objectives:
– Use sum and difference formulas to evaluate
trigonometric functions
– Use sum and difference formulas to verify
trigonometric identities
– Use sum and difference formulas to solve
trigonometric equations
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
3
Sum and Difference Formulas:
sin(u  v)  sin u cos v  cos u sin v
Signs are
the same
sin(u  v)  sin u cos v  cos u sin v


Example:
Verify the identity sin   x  cos x.
2
sin   x  sin  cos x  cos  sin x
2
2
2
 (1) cos x  (0) sin x
 cos x


Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
Sum and Difference Formulas (continued):
cos(u  v)  cos u cos v  sin u sin v
Signs are
opposite
cos(u  v)  cos u cos v  sin u sin v
Signs are the same
tan(u  v)  tan u  tan v
1  tan u tan v
Signs are the same
tan(u  v)  tan u  tan v
1  tan u tan v
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Opposite
signs
5
Example:
• Find the exact value for cos75°
6 2
4
5.4 Sum & Difference Formula
You Try
• Find the exact value of sin(/12)
6 2
4
5.4 Sum & Difference Formula
Using the formulas to Verify/Simplify
• Verify the co-function identity:


cos  x sin x
2





cos  x cos cos x  sin sin x
2

2
2
 (0)(cosx)  (1)(sin x)
 sin x

5.4 Sum & Difference Formula
Evaluating a Trig Expression
Find the exact value of sin(u+v)
given sin u  4 , where 0  u  
5
2

12
and cosv   13 where  v  
2
33

65
5.4 Sum & Difference Formula
Evaluating a Trig Expression
Find the exact value of cos(u-v)
3
15
given cos u   , where   u 
2
17
and
4
sin v 
5

where 0  v  2
77

85
5.4 Sum & Difference Formula
Simplifying Expressions
3
Simplify the expression: cos(  )
2
 sin 
5.4 Sum & Difference Formula
Solving a Trig Equation


Find all solutions of sin( x  )  sin( x  )  1
4
4
in the interval [0,2 )
5
7
x
, x
4
4
5.4 Sum & Difference Formula
Graphical Solution


Find all solutions of sin( x  )  sin( x  )  1
4
4
in the interval [0,2 )
5
7
x
, x
4
4
5.4 Sum & Difference Formula
Solving a Trig Equation
Find all solutions of tan( x   )  2sin( x   )  0
in the interval [0,2 )
x  0,  ,
 5
3
,
3
5.4 Sum & Difference Formula
Homework
5.4 pg.372: 1-5 odd, 15,19, 23-29 odd, 35-41odd,
47-55 odd
Quiz tomorrow (Friday) on verifying trig
identities and solving trig equations
(Sections 5.1-3)
5.4 Sum & Difference Formula