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MA 154
Section 7.3
Lesson 14
Delworth
The Addition and Subtraction Formulas
We refer to the sine and cosine functions as cofunctions of each other. Similarly, the tangent and
cotangent functions are cofunctions, as are the secant and cosecant. If u is the radian measure of


an acute angle, then the angle with radian measure   u is complementary to u.
2

If u is the degree measure of an acute angle, then the angle with degree measure 90  u is
complementary to u.
Cofunction Formulas:
If u is a real number or the radian measure of an angle,
then




(1) cos  u sin u
(2) sin  u cos u
2

2





(3) tan  u cot u
(4) cot   u tan u
2

2 




(5) sec  u csc u
(6) csc  u sec u
2

2

If u is the degree measure of an angle, then
(1) cos90 u  sin u
(2) sin 90 u  cos u
(3) tan90  u  cot u
(4) cot 90  u  tan u
(5) sec90  u  csc u
(6) csc90  u  sec u
Express as a cofunction of a complementary angle.
tan(23.54)
sin(1/6)

cos(/5)
csc(0.74)
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MA 154
Lesson 14
Section 7.3
Delworth
The Addition and Subtraction Formulas
We will now talk about the formulas that involve trigonometric functions of u + v or u – v for any real
numbers or angles u and v. These formulas are known as addition and subtraction formulas,
respectively, or as sum and difference identities.
Addition and subtraction formulas for Cosine, Sine and Tangent
sin( u  v)  sin ucosv  cosu sin v
sin( u  v)  sin ucosv  cosusin v
cos(u  v)  cosucos v sin usin v
cos(u  v)  cosucos v sin usin v
tan(u  v) 
tan u  tan v
1 tan utan v
tan(u  v) 
tan u  tan v
1 tan utan v
Find the exact values.
a)
sin(2/3) + sin(/4)
b)
sin(11/12)
(Hint: Use 11/12 = 2/3 + /4)
a)
tan(30) + tan (225)
b)
tan(255)
(Hint: Use 225 = 30 +225)
a)
cos(/3) – cos(/4)
b)
cos(/12)
(Hint: Use /12 = /3 – /4)
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MA 154
Section 7.3
Lesson 14
Delworth
The Addition and Subtraction Formulas
Express as a trigonometric function of one angle.
cos17cos25  sin17sin 25
sin 25cos17  cos25sin17
Using addition formulas to find the quadrant containing an angle.
If  and  are acute angles such that tan  
15
12
and sin   , find
8
13
a) sin    
b) cos    
c) The quadrant containing  + 
5
24
If  and  are acute angles such that csc   and cot  
, find
3
7
a) sin    
b) tan    
c) The quadrant containing  + 

3
MA 154
Lesson 14
Section 7.3
Delworth
The Addition and Subtraction Formulas
Using addition formulas to find the quadrant containing an angle.
If tan  
15
13
and sec   
for a third-quadrant angle  and second-quadrant angle , find
8
12
a) sin    
b) tan    
c) The quadrant containing  + 

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MA 154
Section 7.3
Lesson 14
Delworth
The Addition and Subtraction Formulas
If  and  are fourth-quadrant angles such that sec  
5
24
and cot    , find
4
7
b) sin    
a) cos   
c) The quadrant containing  - 
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MA 154
Section 7.3
Lesson 14
The Addition and Subtraction Formulas
Verify the identity:
   2
cos  
cos  sin  
 4  2
   tan x  1
tanx  
 4  tan x  1
cos(h  t)  cos(h  t)  2coshcost
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Delworth