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TRIG-Fall 2011-Jordan
Trigonometry, 9th edition, Lial/Hornsby/Schneider, Pearson, 2009
Chapter 6: Inverse Circular Functions and Trigonometric Equations
Section 6.1
Inverse Circular Functions
Inverse Functions
A function must be one-to-one in order to have an inverse function.
A function is one-to-one if it passes the horizontal line test.
None of the six trigonometric functions are one-to-one, so we must restrict the domains
of the functions in order to get the functions to be one-to-one (and, therefore, have
inverse functions).
Inverse Sine Function
Restrict the domain of the sine function to 

2
x

2
-1
y = sin x if and only if sin y = x
where -1 ≤ x ≤ 1 and –π/2 ≤ y ≤ π/2
y = arcsin x ↔ y = sin-1 x
Inverse Cosine Function
Restrict the domain of the cosine function to 0  x  
y = cos-1 x if and only if cos y = x
where -1 ≤ x ≤ 1 and 0 ≤ y ≤ π
y = arccos x ↔ y = cos-1 x
Inverse Tangent Function
Restrict the domain of the tangent function to 
y = tan-1 x if and only if tan y = x
where - ≤ x ≤  and –π/2 < y < π/2
y = arctan x ↔ y = tan-1 x

2
x

2
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Other Inverse Trigonometric Functions
You will be expected to memorize the domain and range for the inverse functions of
sine, cosine, tangent, and cotangent.
Example 1
a)
Find the exact value of each real number y. Do not use a calculator.

2

y  cos 1  

2


b) y = arcsin (-1/2)
c)
y  tan 1 3
d) y = cos -1 1
e)
2 3

y  arc sec 

3


Example 2
Find the degree measure of θ. Do not use a calculator.
θ = csc -1 (-2)
Example 3
Use a calculator to find the real number value: y = arctan (-6.24)
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Compositions of Trigonometric Functions and their Inverses
Start with the innermost function and either evaluate it directly or make a substitution for
it.
Then evaluate the outer function.
Remember to keep in mind the restricted domains and ranges of the inverse functions.
Example 4
Find the exact value of the given expression.
a) cos (cos-1 2)
b) cos (sin-1 (5/13))
c) sin (sin-1 2/3 + cos-1 ½)
Example 5
Write the following expression as an algebraic (non-trigonometric)
expression in u, u > 0:
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Section 6.2
Trigonometric Equations I
Techniques Used in Solving Trigonometric Equations
If only one trigonometric function is present, solve the equation for that function.
If more than one trigonometric function is present:
Re-arrange the equation so that one side equals 0. Then try to factor and set
each factor equal to 0 to solve.
OR
You may need to use an identity to change the form of the equation.
If the equation is quadratic in form, but not factorable, use the quadratic formula. Check
that solutions are in the desired interval.
If the argument is x, the answer should be in radians. If the argument is , the answer
should be in degrees.
One Trigonometric Function Present
Example 1
Solve
for exact solutions over the interval [0, 2π).
Factoring
Example 2
Solve
for exact solutions over the interval [0, 2π).
Example 3
Solve
for exact solutions over the interval [0°, 360°).
Example 4
Solve
. Round to the nearest tenth as appropriate.
Quadratic Formula:
Example 5 Solve
nearest tenth as appropriate.
over the interval [0°, 360°). Round to the
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Section 6.3
Trigonometric Equations II
Solving Trigonometric Equations Involving Functions of Half-Angles and
Multiple Angles
You may have to adjust the solution intervals to fit the given domains.
If a mixture of different types of angles are within an equation, try to use an identity to
rewrite part of the equation.
Solve for exact solutions over the interval [0, 2π).
Example 1
Example 2
Example 3
cos 2x – 2 cos x + 1 = 0
Solve for exact solutions over the interval [0°, 360°).
Example 4
2 sin θ = 2 cos 2θ
Give all exact solutions (in degrees).
Example 5
sin 2θ – sin θ = 0