2. CHAPTER 7: Trigonometric Identities, Inverse Functions, and Equations
7.1 Identities: Pythagorean and Sum and Difference
7.2 Identities: Cofunction, Double-Angle, and Half-Angle
7.3 Proving Trigonometric Identities
7.4 Inverses of the Trigonometric Functions
7.5 Solving Trigonometric Equations
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3. 7.5 Solving Trigonometric Equations
Solve trigonometric equations.
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4. Solve Trigonometric Equations
When an equation contains a trigonometric expression with a variable, such as cos x, it is called a trigonometric equation. Some trigonometric equations are identities, such as sin2 x + cos2 x = 1. Now we consider equations, such as 2 cos x = –1, that are usually not identities. As we have done for other types of equations, we will solve such equations by finding all values for x that make the equation true.
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5. Copyright © 2009 Pearson Education, Inc.
Example
Solve
Solution:
First solve for cos x.
2π/3 and 4π/3 have cosine –1/2.
These numbers, plus any multiple of 2π are the solutions.
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6. Copyright © 2009 Pearson Education, Inc.
Example
Solve
Solution:
First solve for tan 2x.
We are looking for solutions x to the equation for which ≤ x < 2π.
Multiplying by 2, we get 0 ≤ 2x < 4π which is the interval we use when solving tan 2x = –1.
Using the unit circle, find the points 2x in [0, 4π) for which tan 2x = –1.
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7. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
They are:
The values of x, are found by dividing each of these by 2.
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8. Copyright © 2009 Pearson Education, Inc.
Example
Solve
Solution:
Use a calculator in DEGREE mode to find the reference angle = cos– ≈ 65.06º.
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9. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Since cos  is positive, the solutions are in quadrants I and IV. The solutions in [0º, 360º) are
65.06º and 360º – 65.06º = º
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10. Copyright © 2009 Pearson Education, Inc.
Example
Solve
Solution:
Use the principal of zero products:
The solutions in [0º, 360º) are 60º, 180º and 300º.
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11. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Graphical Solution: INTERSECT METHOD
Graph the equations:
and use the INTERSECT feature on the calculator.
The left most solution is 60º. Use the INTERSECT feature two more times to find the solutions, 180º and 300º.
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12. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Graphical Solution: ZERO METHOD
Write the equation in the form
Then graph
The left most zero is 60º. Use the ZERO feature two more times to find the solutions, 180º and 300º.
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13. Copyright © 2009 Pearson Education, Inc.
Example
Solve
Solution:
Use the quadratic formula: a = 10, b = –12, and c = –7.
No solution.
reference angle: 25.44º
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14. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
Sin x is negative, the solutions are in quadrants III & IV.
The solutions are 180º º = º and 300º – 25.44º = º.
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15. Copyright © 2009 Pearson Education, Inc.
Example
Solve each of the following in [0, 2π).
Solution:
We cannot find the solutions algebraically. We can approximate them with a graphing calculator.
a. On the screen on the next slide, on the left side we use the INTERSECT METHOD. Graph
On the screen on the right side we use the ZERO METHOD. Graph
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16. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
The solution in [0, 2π) is approximately 1.32.
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17. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
We cannot find the solutions algebraically. We can approximate them with a graphing calculator.
On the screen on the next slide, on the left side we use the INTERSECT METHOD. Graph
On the screen on the right side we use the ZERO METHOD. Graph
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18. Copyright © 2009 Pearson Education, Inc.
Example
Solution continued:
The solutions in [0, 2π) are approximately 1.13 and 5.66.
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