Transcript 6.3 Notes

6
Inverse
Circular
Functions and
Trigonometric
Equations
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Inverse Circular Functions
6 and Trigonometric Equations
6.1 Inverse Circular Functions
6.2 Trigonometric Equations I
6.3 Trigonometric Equations II
6.4 Equations Involving Inverse
Trigonometric Functions
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6.3 Trigonometric Equations II
Equations with Half-Angles ▪ Equations with Multiple Angles
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Example 1
SOLVING AN EQUATION WITH A HALFANGLE
(a) over the interval
and
(b) for all solutions.
The two numbers over the interval
value
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with sine
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Example 2
SOLVING AN EQUATION USING A
DOUBLE ANGLE IDENTITY
Factor.
or
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Caution
In Example 2, because 2 is not a factor
of cos 2x,
The only way to change cos 2x to a
trigonometric function of x is by using
one of the identities for cos 2x.
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Example 3a
SOLVING AN EQUATION USING A
MULTIPLE-ANGLE IDENTITY
From the given interval 0° ≤ θ < 360°, the interval for
2θ is 0° ≤ 2θ < 720°.
Solution set: {30°, 60°, 210°, 240°}
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Example 3b
Solve
SOLVING AN EQUATION USING A
MULTIPLE-ANGLE IDENTITY
for all solutions.
All angles 2θ that are solutions of the equation
are found by adding integer multiples of 360° to the
basic solution angles, 60° and 120°.
2  60  360n and 2  120  360n
  30  180n and   60  180n
Solution set, where 180º represents the period of sin2θ:
{30° + 180°n, 60° + 180°n, where n is any integer}
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Example 4
SOLVING AN EQUATION WITH A
MULTIPLE ANGLE
Solve tan 3x + sec 3x = 2 over the interval
One way to begin is to express everything in terms of
secant.
Square both sides.
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Example 4
SOLVING AN EQUATION WITH A
MULTIPLE ANGLE (continued)
Multiply each term of the inequality
find the interval for 3x:
by 3 to
Using a calculator and the fact that cosine is positive
in quadrants I and IV, we have
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Example 4
SOLVING AN EQUATION WITH A
MULTIPLE ANGLE (continued)
Since the solution was found by squaring both sides
of an equation, we must check that each proposed
solution is a solution of the original equation.
Solution set: {0.2145, 2.3089, 4.4033}
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