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Math Analysis
Chapter 5 Notes: Analytic Trigonometric
Day 12: Section 5.1-Verifying Trigonometric Identities
Fundamental Trig Identities
Reciprocal Identities:
1
1
sin u 
cos u 
csc u
sec u
csc u 
1
sin u
sec u 
Quotient Identities:
sin u
tan u 
cos u
1
cos u
cot u 
tan u 
1
cot u
cot u 
1
tan u
cos u
sin u
Pythagorean Identities:
sin 2 u  cos2 u  1
1  tan 2 u  sec2 u
1  cot 2 u  csc2 u
Even-Odd Identities:
sin( u )   sin u
csc( u )   csc u
cos( u )  cos u
sec(u )  sec u
tan( u )   tan u
cot( u )   cot u
Practice: Use the given value to evaluate the remaining trig functions.
2
1. sin x  , tan x  0
3
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Practice: In 2-5, Write each expression as a single fraction. Reduce, if possible.
sin 
cos 
1
1


2.
3.
1  cos  sin 
csc x  1 csc x  1
4. csc x 
cot 2 x
csc x
Practice: In 6-11, Verify the identity algebraically.
cos x
sin x

 sec x
6.
1  sin x cos x
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5.
sin x
1  cos x

1  cos x
sin x
7.
sin x
 cos x
tan x
2
8. sin   cos cot   csc
10.
cos x sin x

1
sec x csc x
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9. cos 2 x  sin 2 x  2 cos 2 x  1
11.
sec   cos 
 sin 2 
sec 
3
Day 13: Section 5-2: Sum and Difference Formulas:
After today’s lesson you should be able to find the exact value of and angle using the sum and difference formulas.
The Sum Formulas: Given two angles θ and β:


sin(   )  sin  cos   cos sin 

tan(   ) 
The Difference Formulas: Given two angles θ and β:
cos(   )  cos  cos   sin  sin 
sin(   )
cos(   )
or tan(   ) 
tan   tan 
1  tan  tan 


sin(   )  sin  cos   cos  sin 

tan(   ) 
cos(   )  cos  cos   sin  sin 
sin(   )
cos(   )
or tan(   ) 
tan   tan 
1  tan  tan 
Practice: Use the correct formula to evaluate the given trigonometric expression.

5
2. cos   
3 6 
1. sin (450 – 300)
3. Find all three trig functions of the given angle:  
13
12
using the sum or difference formula.
4. Write the equivalent expression as sine, cosine, or tangent of an angle. Then find the exact value of the
expression.
sin
5

5

cos  cos sin
12
4
12
4
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5. Verify the trig identity: sin2x = 2sinxcosx
6. Verify the trig identity:
sin( x  y)
 tan x  tan y
cos x cos y
7. Find the exact value of the of the following under the given condition:
(a) cos(x + y)
3
5
(b) sin(x + y)
Given: sin x  , and x lies in Quad II, and cos y 
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(c) tan (x + y)
5
, and y lies in Quad IV.
13
5
Day 14: Section 5-3 Half Angle Formulas and Double Angle Formulas:
Half Angle Formulas:
u
1
cos   1  cos u 
2
2
tan
u
sin u

2 1  cos u
sin
u
1
  1  cos u 
2
2
tan
u 1  cos u

2
sin u
u
u
u
The signs of sin   and cos   depend on the quadrant in which
lies.
2
2
2
u
u
u
Practice: Calculate sin , cos , and tan for the angle described.
2
2
2
5 3
 u  2
1. cos u  ;
13 2
3
3
2. sin u   ;   u 
4
2
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3. cos u  
5 
;
u 
13 2
Practice: Use the half-angle formulas to determine the exact values of the sine, cosine, and
tangent of the angle.
1. 195
2.
9
8
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3.
19
12
4. 105
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Double Angle Formulas:
sin 2u  2sin u cos u
cos 2u  cos 2 u  sin 2 u
2 tan u
1  tan 2 u
Practice: Find the exact value of the expression


2 tan1050
1.
2. cos 2  sin 2
2
0
12
12
1  tan 105
tan 2u 
Practice: If sinθ =
3. 2sin
7
7
cos
12
12
4
and θ lies in quadrant II, find the exact value of each of the following
5
4. sin2θ
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5. cos2θ
6. tan2θ
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Day 15 Section 5-5 Trigonometric Equations
Tips for Solving Trig Equations
1. Get an equation (or equations) in which one trig function equals a constant
2. Substitute any obvious trig identities
3. Reduce the number of different trig functions (get the equation so it only has 1 trig function).
4. Do any obvious algebra (get terms into 1 fraction, factor, use quadratic formula). Remember, in order to
factor the equation MUST be equal to 0!!!
5. Do not divide both members by a variable (you can NOT divide by a trig function).
Always find general solution first!!! This is especially important if the angle (x) has a coefficient!!!
Examples: Solve the equation (a) giving the general solutions and (b) in the interval [0, 2π).
3 sec x  2  0
1. 2 cos x  1  0
2.
3. 3 csc 2 x  4  0
4. 2 sin 2 2 x  1
5. 2 cos 3x  1  0
6. 2 cos
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x
 2
2
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Day 16 Section 5-5 Trigonometric Equations with Factoring
Examples: Solve the equation (a) giving the general solutions and (b) in the interval [0, 2π).
1.  3 tan 2 x  1 tan 2 x  3  0
2. tan2 x  tan x  0
3. 2sin2 x  sin x 1  0
4. 2 cos x sin x  cos x  0
5. sin 2 x  sin x


6. sin  x    sin  x    1
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
4

4
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Chapter 5 Review
For questions 1-4, solve each equation in the domain  0, 2  .
1. 2sin 2 2 x  1
2. 3 tan 4 x  10 tan 2 x  3  0
3. 2cos3x 1  0
4. 2cos x  sin 2x  0
For questions 5-7, find the exact value of each expression using the sum and difference of angles
formulas.
11
5. cos 255 
6. tan
12
7. sin
7
12
For questions 8-13, given that sin u  
8. cos 2u
10. tan 2u
12. tan
u
2
12 3
,
 u  2 , find the value of the following expressions.
13 2
u
9. sin
2
11. sin 2u
13. cos
u
2
For questions 14-16, use the half-angle formulas to determine the exact values of the sine, cosine, and
tangent of the angle.
5
14. tan195
15. sin
12
16. cos
7
8
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