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Inverse Trigonometric Functions
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Inverse Trigonometric Functions
4 Activities
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f(x) = \frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
Domain:
[
−
1
,
0
)
∪
(
0
,
1
]
;
Range:
(
−
∞
,
−
2
/
π
]
∪
[
2
/
π
,
∞
)
\text{Domain: } [-1,0) \cup (0,1] ; \text{ Range: } (-\infty, -2/\pi]\cup[2/\pi, \infty)
Domain:
[
−
1
,
0
)
∪
(
0
,
1
]
;
Range:
(
−
∞
,
−
2/
π
]
∪
[
2/
π
,
∞
)
Domain:
[
−
1
,
0
)
∪
(
0
,
1
]
; Range:
[
−
2
/
π
,
0
)
∪
(
0
,
2
/
π
]
\text{ Domain: } [-1,0) \cup (0,1] \text{; Range: } [-2/\pi,0) \cup (0,2/\pi]
Domain:
[
−
1
,
0
)
∪
(
0
,
1
]
; Range:
[
−
2/
π
,
0
)
∪
(
0
,
2/
π
]
Domain:
(
−
1
,
1
)
; Range:
(
−
2
/
π
,
2
/
π
)
\text{ Domain: } (-1,1) \text{; Range: } (-2/\pi, 2/\pi)
Domain:
(
−
1
,
1
)
; Range:
(
−
2/
π
,
2/
π
)
Domain:
(
−
1
,
1
)
; Range:
(
−
2
/
π
,
0
)
∪
(
0
,
2
/
π
)
\text{ Domain: } (-1,1) \text{; Range: } (-2/\pi, 0) \cup (0, 2/\pi)
Domain:
(
−
1
,
1
)
; Range:
(
−
2/
π
,
0
)
∪
(
0
,
2/
π
)
Domain:
[
−
1
,
1
]
; Range:
[
−
2
/
π
,
2
/
π
]
\text{ Domain: } [-1,1] \text{; Range: } [-2/\pi, 2/\pi]
Domain:
[
−
1
,
1
]
; Range:
[
−
2/
π
,
2/
π
]
I don't know
Check Submission
More Inverse Trigonometric Functions Questions:
Inverse Trigonometric Functions
Compute the exact value of
arccos
(
cos
(
3
π
4
)
)
\arccos \left ( \cos \left ( \frac{3 \pi}{4} \right ) \right )
arccos
(
cos
(
4
3
π
)
)
Find the exact value of
arcsin
(
sin
(
−
17
π
6
)
)
\arcsin(\sin(-\frac{17\pi}{6}))
arcsin
(
sin
(
−
6
17
π
))
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Domain & Range with Inverse Trig
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f\left(x\right)=\frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Inverse Trigonometric Functions
Evaluate
tan
(
tan
−
1
π
4
)
+
arcsin
(
sin
5
π
4
)
\tan\left(\tan^{-1}\frac{\pi}{4}\right)+\arcsin\left(\sin\frac{5\pi}{4}\right)
tan
(
tan
−
1
4
π
)
+
arcsin
(
sin
4
5
π
)
.
Inverse Trigonometric Functions
Find the exact value of
cos
[
tan
−
1
2
+
tan
−
1
3
]
\cos[\tan^{-1}2+\tan^{-1}3]
cos
[
tan
−
1
2
+
tan
−
1
3
]
.
Inverse Trig with Product
Find the exact value of
sin
[
2
cos
−
1
(
1
5
)
]
\sin\bigg[2\cos^{-1}\bigg(\dfrac{1}{5}\bigg)\bigg]
sin
[
2
cos
−
1
(
5
1
)
]
.
Inverse Trigonometric Functions: Restrict the Domain
Find the exact value of
arccos
(
cos
23
π
3
)
\arccos\bigg(\cos\dfrac{23\pi}{3}\bigg)
arccos
(
cos
3
23
π
)
.
Inverse Trigonometric Functions
Simplify
cos
(
arctan
(
4
)
)
\cos(\arctan(4))
cos
(
arctan
(
4
))
.
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Inverse Trigonometric Functions
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Inverse Trigonometric Functions
Without the using a calculator, compute the exact value of
arctan
(
tan
(
13
π
11
)
)
\arctan \left ( \tan \left ( \frac{13\pi}{11} \right ) \right )
arctan
(
tan
(
11
13
π
)
)
*The
π
\pi
π
symbol is obtained by typing in \pi
Inverse Trigonometric Functions
Answer the following Trig and Inverse Trig questions.
Inverse Trigonometric Functions
Without the use a calculator, compute the exact value of
arccos
(
cos
(
3
π
4
)
)
\arccos \left ( \cos \left ( \frac{3 \pi}{4} \right ) \right )
arccos
(
cos
(
4
3
π
)
)
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Inverse Trigonometric Functions
Find the exact value of
sin
(
2
cos
−
1
(
x
)
)
\sin\left(2\cos^{-1}\left(x\right)\right)
sin
(
2
cos
−
1
(
x
)
)
.
Inverse Trigonometric Functions
Find the value of sin (arccos (− √3/2 )).
Inverse Trigonometric Functions
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Inverse Trigonometric Functions
Simplify the following expression:
sec
2
(
arctan
(
1
/
x
2
)
)
\sec^2(\arctan(1/x^2))
sec
2
(
arctan
(
1/
x
2
))
Inverse Trigonometric Functions
Simplify
sin
(
2
arcsin
x
)
\sin(2\arcsin x)
sin
(
2
arcsin
x
)
Inverse Trigonometric Functions
Find the exact value of
tan
[
arcsin
(
−
3
/
5
)
]
\tan[\arcsin(-3/5)]
tan
[
arcsin
(
−
3/5
)]
Inverse Trigonometric Functions
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin(2\cos^{-1}x)
sin
(
2
cos
−
1
x
)
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for
x
:
a.
9
x
−
2
=
(
1
27
)
2
x
+
1
9^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
9
x
−
2
=
(
27
1
)
2
x
+
1
Simplify the following expression:
sin
[
−
tan
−
1
(
x
2
−
1
)
]
\sin[-\tan^{-1}(\sqrt{x^2-1})]
sin
[
−
tan
−
1
(
x
2
−
1
)]
Practice: Trigonometric Modeling
Q
:
\bf{Q:}
Q
:
The temperature of the body oscillates throughout the day and can be modeled by the function:
T
(
t
)
=
37
+
0.4
cos
(
π
12
t
−
π
3
)
T\left(t\right)=37\ +\ 0.4\cos\left(\dfrac{\pi}{12} \ t-\dfrac{\pi}{3}\right)
T
(
t
)
=
37
+
0.4
cos
(
12
π
t
−
3
π
)
where
T
T
T
is the temperature in degrees Celsius, and
t
t
t
is the time in hours after 12pm.
a) What is the maximum and minimum temperatures of the body throughout the day?
b) What is the period of this function? Does this make sense given the context of the problem?
Inverse Trigonometric Functions: Restrict the Domain
Find the exact value of
arccos
(
cos
23
π
3
)
\arccos\bigg(\cos\dfrac{23\pi}{3}\bigg)
arccos
(
cos
3
23
π
)
.
Inverse Trigonometric Functions
Find the exact value of
cos
[
tan
−
1
2
+
tan
−
1
3
]
\cos[\tan^{-1}2+\tan^{-1}3]
cos
[
tan
−
1
2
+
tan
−
1
3
]
.
Inverse Trig with Product
Find the exact value of
sin
[
2
cos
−
1
(
1
5
)
]
\sin\bigg[2\cos^{-1}\bigg(\dfrac{1}{5}\bigg)\bigg]
sin
[
2
cos
−
1
(
5
1
)
]
.
Inverse Trigonometric Functions
Simplify the expression
cos
(
tan
−
1
(
x
2
)
)
\cos\left(\tan^{-1}\left(\frac{x}{2}\right)\right)
cos
(
tan
−
1
(
2
x
)
)
Inverse Trigonometric Functions
Simplify the following expression;
sec
−
1
(
2
)
=
\sec^{-1}(2)=
sec
−
1
(
2
)
=
Inverse Trigonometric Functions
Calculate the value of the expression
sin
−
1
(
sin
(
5
π
3
)
)
\sin^{-1}\left(\sin\left(\frac{5\pi}{3}\right)\right)
sin
−
1
(
sin
(
3
5
π
)
)
Find the exact value of
arcsin
(
sin
(
−
17
π
6
)
)
\arcsin(\sin(-\frac{17\pi}{6}))
arcsin
(
sin
(
−
6
17
π
))
Inverse Trigonometric Functions
Find
sin
(
arctan
(
x
)
)
\sin\left(\arctan\left(x\right)\right)
sin
(
arctan
(
x
)
)
.
Practice: Trig & Inverse Trig Values (level 2)
Practice: Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Practice: Domain & Range with Inverse Trig
Practice: Domain & Range with Inverse Trig
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f\left(x\right)=\frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
.
Find the value of
sin
[
arccos
(
−
3
2
)
]
\sin\left[\arccos\left(-\frac{\sqrt{3}}{2}\right)\right]
sin
[
arccos
(
−
2
3
)
]
.
Inverse Trigonometric Functions
Find sin (arctan 𝑥).
Trig & Inverse Trig Values
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.
Trig & Inverse Trig Values
Find the exact value of
sin
(
2
cos
−
1
x
)
\sin\left(2\cos^{-1}x\right)
sin
(
2
cos
−
1
x
)
.
Domain & Range with Inverse Trig
Find the domain and range of
f
(
x
)
=
1
arcsin
x
f\left(x\right)=\frac{1}{\arcsin x}
f
(
x
)
=
a
r
c
s
i
n
x
1
.
Inverse trigonometric functions
Evaluate the expression
arcsin
(
−
1
2
)
×
cos
(
7
π
6
)
+
2
ln
(
3
e
)
\arcsin\left(-\frac{1}{\sqrt{2}}\right)\times\cos\left(\frac{7\pi}{6}\right)+2\ln\left(3\sqrt{e}\right)
arcsin
(
−
2
1
)
×
cos
(
6
7
π
)
+
2
ln
(
3
e
)
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Inverse Trigonometric Functions
Evaluate
tan
(
tan
−
1
π
4
)
+
arcsin
(
sin
5
π
4
)
\tan\left(\tan^{-1}\frac{\pi}{4}\right)+\arcsin\left(\sin\frac{5\pi}{4}\right)
tan
(
tan
−
1
4
π
)
+
arcsin
(
sin
4
5
π
)
.
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of
tan
[
arcsin
(
−
3
5
)
]
\tan\left[\arcsin\left(-\frac{3}{5}\right)\right]
tan
[
arcsin
(
−
5
3
)
]
.