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SESSION 6
Trig. Equations and Identities
Math 30-1
3
R
(Revisit, Review and Revive)
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Mathematics 30-1 Learning Outcomes
Specific Outcome 5: Solve, algebraically and graphically, first and second degree
trigonometric equations with the domain expressed in degrees and radians.
- Verify, with or without technology, that a given value is a solution to a
trigonometric equation.
- Determine, algebraically, the solution of a trigonometric equation, stating the
solution in exact form when possible.
- Determine, using technology, the approximate solution of a trigonometric
equation in a restricted domain
- Relate the general solution of a trigonometric equation to the zeros of the
corresponding trigonometric function (restricted to sine and cosine functions).
- Determine, using technology, the general solution of a given trigonometric
equation
- Identify and correct errors in a solution for a trigonometric equation.
Key ideas:
Verify means to see if it works for a certain value of the variable. If you put in a value
and its false then it cannot be an identity. If you put in a value and it is true then it
may be an identity. You may also verify a trig identity graphically in your calculator
(both graphs should appear the same). Verifying an equation is not enough to conclude
that the equation is an identity.
Prove: means to show that the left side is equal to the right side for all values of the
variable in the domain. When proving you may not move anything from one side of
the equation to the other. You have to work with each side of the equation
independently.
- Try to simplify the more complicated side
- Use identities from your formula sheet to make substitutions
- Rewrite in terms of sine and cosine only
- factor
Solve: find the particular value of the variable for which the equation is true. When
solving you may do the same thing to both sides of the equation to isolate the variable.
- Isolate trig ratio
- Factor with one side of the equation being equal to zero
- Simplify using trig identities then solve
- Remember to check for non-permissible values.
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7
is a solution to the trigonometric equation
6
3
.
5sin   2  1  3sin  in the domain    
2
Example 1: Verify that  
Example 2: Determine the exact roots for the trigonometric equation
2cos   3  0 in the domain 0    2 .
Example 3: Solve the trigonometric equation in the specified domain.
2sec  6  0, 0    360
Example 4: How is the general solution of 2cos2   3cos   1  0 related to the zeros of
y  2cos2   3cos   1 ?
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Example 5: Determine the general solution for 3csc  6  0 .
Example 6: Solve the following trigonometric equation.
9sin 2   12sin   4  0,   0, 360 
Practice Questions:
1.
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2.
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3.
4.
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5.
6.
2.
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8.
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Specific Outcome 6: Prove trigonometric identities, using:
• reciprocal identities
• quotient identities
• Pythagorean identities
• sum or difference identities (restricted to sine, cosine and tangent)
• double-angle identities (restricted to sine, cosine and tangent).
-
Explain the difference between a trigonometric identity and a trigonometric
equation
Verify a trigonometric identity numerically for a given value in either degrees
or radians.
Explain why verifying that the two sides of a trigonometric identity are equal
for given values is insufficient to conclude that the identity is valid.
Determine, graphically, the potential validity of a trigonometric identity, using
technology.
Determine the non-permissible values of a trigonometric identity
Prove, algebraically, that a trigonometric identity is valid.
Determine, using the sum, difference and double-angle identities, the exact
value of a trigonometric ratio.
Key ideas:
trigonometric equation
An equation involving trigonometric ratios.
trigonometric identity
A trigonometric equation that is true for all permissible
values of the variable in the expression on both sides of the equation.
Note: Sometimes the symbol  is used instead of = to indicate that a statement is an
identity.
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Examples:
Example 7: Verify that the equation
  30 and for  

4
sec 
 sin  is true for
tan   cot 
.
Example 8: Explain why verifying that the two sides of a trigonometric identity are equal
for given values is insufficient to conclude that the identity is valid.
sin  cos  1  cos 
. Graph the two sides of the

1  cos 
tan 
equation using technology, over the domain 0    2 . Could it be an identity?
Example 9: Consider the equation
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1  cos 
sin 
. What are the non-permissible

sin 
1  cos 
values for the identity over the domain 0    360 .
Example 10: Consider the identity
Example 11: Prove each identity.
a)
cos   cot 
 cos  cot 
sec   tan 
b) sec  tan  
cos 
1  sin 
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c) Prove the identity sin2x= 2sinx cosx
Example 12: Determine the exact value for each expression.
a) sin

12
b) cos 75
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Practice:
9.
10.
11.
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12.
13.
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14.
15.
16.
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17.
18.
19.
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20.
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Practice Questions Answers
1. C
2. D
3. D
4. A
5. B
6. C
7. A
8. B
9. C
10. A
11. D
12. A
13. C
14. A
15. C
16. D
17. D
18. A
19. C
20. C
N.R.
1. 1.50
2. 0.7
3. 1
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