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AP Precalculus
Unit 3: Trigonometric and Polar Functions
3.3 Using inverse trigonometric functions to solve trigonometric equations
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Cards (55)
What do inverse trigonometric functions take as input?
A ratio
What is the domain of the inverse sine function,
sin
−
1
(
x
)
\sin^{ - 1}(x)
sin
−
1
(
x
)
?
-1 ≤ x ≤ 1
What is the range of the inverse cotangent function,
cot
−
1
(
x
)
\cot^{ - 1}(x)
cot
−
1
(
x
)
?
0
<
θ
<
π
0 < θ < \pi
0
<
θ
<
π
An example of a regular trigonometric function is
s
i
n
(
30
°
)
sin(30°)
s
in
(
30°
)
, which equals 0.5
Match each inverse trigonometric function with its domain:
tan
−
1
(
x
)
\tan^{ - 1}(x)
tan
−
1
(
x
)
↔️ Domain: -∞ < x < ∞
cos
−
1
(
x
)
\cos^{ - 1}(x)
cos
−
1
(
x
)
↔️ Domain: -1 ≤ x ≤ 1
sec
−
1
(
x
)
\sec^{ - 1}(x)
sec
−
1
(
x
)
↔️ Domain: x ≤ -1 or x ≥ 1
What is the range of the inverse tangent function, `tan⁻¹(x)`?
-π/2 < θ < π/2
What is the range of the inverse cotangent function, `cot⁻¹(x)`?
0 < θ < π
Steps to solve trigonometric equations using inverse trigonometric functions
1️⃣ Isolate trigonometric expressions in equations
2️⃣ Apply appropriate inverse trigonometric functions
3️⃣ Check solutions for validity
Match each inverse trigonometric function with its domain and range:
sin
−
1
(
x
)
\sin^{ - 1}(x)
sin
−
1
(
x
)
↔️ Domain: -1 ≤ x ≤ 1, Range:
−
π
2
≤
θ
≤
π
2
- \frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}
−
2
π
≤
θ
≤
2
π
cos
−
1
(
x
)
\cos^{ - 1}(x)
cos
−
1
(
x
)
↔️ Domain: -1 ≤ x ≤ 1, Range:
0
≤
θ
≤
π
0 ≤ θ ≤ \pi
0
≤
θ
≤
π
tan
−
1
(
x
)
\tan^{ - 1}(x)
tan
−
1
(
x
)
↔️ Domain: -∞ < x < ∞, Range:
−
π
2
<
θ
<
π
2
- \frac{\pi}{2} < θ < \frac{\pi}{2}
−
2
π
<
θ
<
2
π
cot
−
1
(
x
)
\cot^{ - 1}(x)
cot
−
1
(
x
)
↔️ Domain: -∞ < x < ∞, Range:
0
<
θ
<
π
0 < θ < \pi
0
<
θ
<
π
The domain of
tan
−
1
(
x
)
\tan^{ - 1}(x)
tan
−
1
(
x
)
is all real numbers.
True
What is the range of the inverse sine function,
sin
−
1
(
x
)
\sin^{ - 1}(x)
sin
−
1
(
x
)
?
−
π
2
≤
θ
≤
π
2
- \frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}
−
2
π
≤
θ
≤
2
π
The range of the inverse cosine function, `cos⁻¹(x)`, is
0 ≤ θ ≤ π
The range of the inverse secant function, `sec⁻¹(x)`, is
0 ≤ θ ≤ π
What is the output of an inverse trigonometric function?
An angle
The domains and ranges of
inverse trigonometric functions
must be considered when solving trigonometric equations.
True
What is the domain of `sin⁻¹(x)`?
-1 ≤ x ≤ 1
The first step in solving trigonometric equations is to isolate the trigonometric
expression
In the equation `3sin(x) = 4 - 2`, the next step is to simplify the right side to `3sin(x) =
2
To solve for `sin(x) = 0.7`, the appropriate inverse function to apply is
sin^-1
`sin(x)` is positive in the first and
second
quadrants.
True
The ranges of inverse trigonometric functions are unlimited.
False
Steps to isolate a trigonometric expression in an equation
1️⃣ Simplify the equation
2️⃣ Use algebraic operations to move non-trigonometric terms
3️⃣ Divide by the coefficient of the trigonometric expression
What is the first step in solving a trigonometric equation using inverse functions?
Isolate the trigonometric expression
What do regular trigonometric functions take as input and return as output?
Angle as input, ratio as output
The range of `cos⁻¹(x)` is
0 ≤ θ ≤ π
Algebraic operations are used to move non-
trigonometric
terms in trigonometric equations.
True
The domain and range of inverse
trigonometric
functions must be considered to ensure solutions are valid.
True
What interval are we considering for solutions in the equation `sin(x) = 0.7`?
[0, 2π]
What is the second solution for `sin(x) = 0.7` in the interval `[0, 2π]`?
2.366
What is the next step after isolating a trigonometric expression in an equation with multiple trigonometric functions?
Apply the inverse function
What type of trigonometric functions are used to solve equations with multiple trigonometric expressions?
Inverse trigonometric functions
What is the isolated trigonometric expression in the equation `2cos(x) - sin(x) = 1`?
cos(x) = (1 + sin(x))/2
Regular trigonometric functions take an angle as input and return a
ratio
The range of the inverse sine function is -\frac{\pi}{2} ≤ θ ≤ \frac{\pi}{2}</latex>
True
What is the domain of the inverse cosine function,
cos
−
1
(
x
)
\cos^{ - 1}(x)
cos
−
1
(
x
)
?
-1 ≤ x ≤ 1
What is the domain of the inverse sine function, `sin⁻¹(x)`?
-1 ≤ x ≤ 1
The domain of `csc⁻¹(x)` is `x ≤ -1` or `x ≥ 1`.
True
The domains of inverse trigonometric functions are defined in terms of
ratios
, not angles.
True
To isolate `sin(x)` in the equation `3sin(x) + 2 = 4`, the final step is to divide both sides by
3
Since `sin(x)` is positive in the first and second quadrants, the second solution for `sin(x) = 0.7` is
π - 0.775
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