Inverse Circular Functions and Trigonometric Equations · 2017. 4. 13. · Inverse Circular...
Transcript of Inverse Circular Functions and Trigonometric Equations · 2017. 4. 13. · Inverse Circular...
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Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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6Inverse Circular Functions and Trigonometric Equations
Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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Copyright © 2017, 2013, 2009 Pearson Education, Inc.
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Equations Involving Inverse Trigonometric Functions
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Solution for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations
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Example 1 SOLVING AN EQUATION FOR A SPECIFIED VARIABLE
Solve y = 3 cos 2x for x, where x is restricted to the interval
Divide by 3.
Definition of arccosine
Multiply by
0, .2π⎡ ⎤
⎢ ⎥⎣ ⎦
Because y = 3 cos 2x for x has period π, the restriction ensures that this function is one-to-one and has a one-to-one relationship.
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Example 2 SOLVING AN EQUATION INVOLVING AN INVERSE TRIGONOMETRIC FUNCTION
CHECK
Solution set: {1}
Divide by 2.
Definition of arcsine
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Example 3 SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS
and for u in quadrant I,
Substitute.
Definition of arccosine
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Example 3 SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS (continued)
Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.
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Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY
Isolate one inverse function on one side of the equation:
The arccosine function yields angles in quadrants I and II, so, by definition,
Definition of arcsine
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Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)
Sine sum identity
From equation (1) and by the definition of the arcsine function,
and u lies in quadrant I.
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Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)
From the triangle, we have
Substitute.
Multiply by 2.
Subtract x.
Square each side.
Distribute, then add 3x2.
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Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)
Take the square root of each side. Choose the positive root because x > 0.
Now check the solution in the original equation.
Divide by 4.