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Trigonometric Functions: Trig Identities
Related Topics
Wize University Calculus 1 Textbook > Pre-Calculus (Review)
Trigonometric Functions
5 Activities
Which of the following equations (if any) is true?
a.
(
sin
π
8
+
cos
π
8
)
2
=
1
\left(\sin\frac{\pi}{8}+\cos\frac{\pi}{8}\right)^2=1
(
sin
8
π
+
cos
8
π
)
2
=
1
b.
(
sin
π
8
)
2
+
(
cos
π
8
)
2
=
1
+
(
π
8
)
2
\left(\sin\frac{\pi}{8}\right)^2+\left(\cos\frac{\pi}{8}\right)^2=1+\left(\frac{\pi}{8}\right)^2
(
sin
8
π
)
2
+
(
cos
8
π
)
2
=
1
+
(
8
π
)
2
c.
cos
π
8
=
1
12
\cos\frac{\pi}{8}=\frac{1}{\sqrt{12}}
cos
8
π
=
12
1
d.
cos
π
4
=
(
cos
π
8
)
2
−
(
sin
π
8
)
2
\cos\frac{\pi}{4}=\left(\cos\frac{\pi}{8}\right)^2-\left(\sin\frac{\pi}{8}\right)^2
cos
4
π
=
(
cos
8
π
)
2
−
(
sin
8
π
)
2
e. None of the above
I don't know
Check Submission
More Trigonometric Functions Questions:
Trigonometric Functions
Find the number of solutions to the equation cos 2𝑥 = cos 𝑥 − 1 in the interval [0,2𝜋].
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Evaluate
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
Additional Practice Problems
Simplify the following expressions:
a.
ln
(
3
e
2
)
+
e
2
ln
3
\ln\left(3e^2\right)+e^{2\ln3}
ln
(
3
e
2
)
+
e
2
l
n
3
Trigonometric Functions
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
.
Trigonometric Functions
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Practice: Trig Equation
Q
:
\bf{Q:}
Q
:
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Q
:
\bf{Q:}
Q
:
Given
cos
θ
=
12
13
\cos\theta=\dfrac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\dfrac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact value of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Practice: Special Angles
Q
:
\bf{Q:}
Q
:
Evaluate
tan
7
π
6
\tan\dfrac{7\pi}{6}
tan
6
7
π
and
sin
15
π
4
\sin\dfrac{15\pi}{4}
sin
4
15
π
.
Q
:
\bf{Q:}
Q
:
Find all trigonometric function values of the angle
4
π
3
\dfrac{4\pi}{3}
3
4
π
Q
:
\bf{Q:}
Q
:
Find the domain of the following function:
f
(
x
)
=
tan
x
f\left(x\right)=\tan\ x
f
(
x
)
=
tan
x
Trigonometric Functions
Determine the number of solutions the equation
3
cos
x
+
sin
2
x
=
0
\sqrt{3}\cos x+\sin2x=0
3
cos
x
+
sin
2
x
=
0
has in the interval
[
0
,
2
π
]
\left[0,2\pi\right]
[
0
,
2
π
]
.
Even or Odd Trigonometric Functions
Determine if the following functions are even, odd, or neither:
f
(
x
)
=
sin
x
+
cos
x
f\left(x\right)=\sin x+\cos x
f
(
x
)
=
sin
x
+
cos
x
and
g
(
x
)
=
sin
x
sec
x
g\left(x\right)=\dfrac{\sin x}{\sec x}
g
(
x
)
=
sec
x
sin
x
Exponential and Trigonometric Functions
Q
:
\bf{Q:}
Q
:
Find the domain of .
f
(
x
)
=
sin
(
3
x
)
e
4
x
−
1
−
tan
(
2
x
)
f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
f
(
x
)
=
e
4
x
−
1
sin
(
3
x
)
−
tan
(
2
x
)
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Trigonometric Functions
Solve the following equation:
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
2\sin x-2\sqrt{3}\cos x=\sqrt{3}\tan x-3
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
Trigonometric Functions
Given
3
cos
x
+
2
sin
2
x
=
3
3\cos x+2\sin^2x=3
3
cos
x
+
2
sin
2
x
=
3
, solve for
x
x
x
in the domain
(
0
,
−
3
π
2
]
\left(0,-\dfrac{3\pi}{2}\right]
(
0
,
−
2
3
π
]
.
Practice: Transforming a Trig Function
Graph the function:
y
=
3
sin
(
2
x
+
π
2
)
+
1
\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1
y
=
3
sin
(
2
x
+
2
π
)
+
1
Trigonometric Functions
The solutions of the equation
tan
2
x
=
sin
2
x
+
cos
2
x
\tan^2x=\sin^2x+\cos^2x
tan
2
x
=
sin
2
x
+
cos
2
x
in the interval
[
0
,
π
]
\left[0,\pi\right]
[
0
,
π
]
are
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Practice: Transforming a Trig Function
Graph the function:
y
=
3
sin
(
2
x
+
π
2
)
+
1
\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1
y
=
3
sin
(
2
x
+
2
π
)
+
1
Trigonometric Functions
A maintenance technician came to a very expensive hotel to install new shower-heads. The only problem is that ,when turned on, the water from the new shower heads varied over time. When the water was first turned on it was 100 degrees Fahrenheit. After being turned on, the temperature cycled between 80 and 120 degrees Fahrenheit. It took exactly 10 minutes from the time the shower was turned on to cycle back to its original temperature of 100 degrees. Let
T
(
t
)
T(t)
T
(
t
)
measure the temperature of water at time t (in minutes). Which of the following could be an equation for
T
(
t
)
T(t)
T
(
t
)
?
Trigonometric Functions
Find the number of solutions to the equation cos 2𝑥 = cos 𝑥 − 1 in the interval [0,2𝜋].
Trigonometric Functions
Find the number of solutions to the equation
cos
(
2
x
)
=
cos
(
x
)
−
1
\cos\left(2x\right)=\cos\left(x\right)-1
cos
(
2
x
)
=
cos
(
x
)
−
1
in the interval
[
0
,
π
]
\left[0,\pi\right]
[
0
,
π
]
.
Greatest Integer Function
Let
f
(
x
)
=
[
[
sin
x
−
2
]
]
f\left(x\right)=\left[\left[\sin x-2\right]\right]
f
(
x
)
=
[
[
sin
x
−
2
]
]
be the greatest integer of
sin
x
−
2
\sin x-2
sin
x
−
2
on the open interval
(
0
,
π
)
\left(0,\ \pi\right)
(
0
,
π
)
.
The range of
f
(
x
)
f(x)
f
(
x
)
is
Trigonometric Functions
Find all solutions to
sin
(
θ
)
=
sin
(
2
θ
)
\sin(\theta)=\sin(2\theta)
sin
(
θ
)
=
sin
(
2
θ
)
on the interval
[
0
,
2
π
)
[0,2\pi)
[
0
,
2
π
)
Trigonometric Functions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x + \frac{\sin 2x \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
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
Practice: Trig Equation
Practice: Trig Equation
Solve the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
.
Practice: Trig Ratios
Practice: Trig Ratios
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
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?
Practice: Trig Equation
Practice: Trig Equation
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Logarithmic and Trigonometric Functions
Solve for x in the equation
ln
(
4
−
4
sin
2
x
)
=
0
\ln\left(4-4\sin^2x\right)=0
ln
(
4
−
4
sin
2
x
)
=
0
Trigonometric Functions
Given
3
cos
x
+
2
sin
2
x
=
3
3\cos x+2\sin^2x=3
3
cos
x
+
2
sin
2
x
=
3
, solve for
x
x
x
in the domain
(
0
,
−
3
π
2
]
\left(0,-\dfrac{3\pi}{2}\right]
(
0
,
−
2
3
π
]
.
Trigonometric Functions
Solve the following equation:
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
2\sin x-2\sqrt{3}\cos x=\sqrt{3}\tan x-3
2
sin
x
−
2
3
cos
x
=
3
tan
x
−
3
Exponential and Trigonometric Functions
Q
:
\bf{Q:}
Q
:
Find the domain of .
f
(
x
)
=
sin
(
3
x
)
e
4
x
−
1
−
tan
(
2
x
)
f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
f
(
x
)
=
e
4
x
−
1
sin
(
3
x
)
−
tan
(
2
x
)
Q
:
\bf{Q:}
Q
:
Find the domain of the following function:
f
(
x
)
=
tan
x
f\left(x\right)=\tan\ x
f
(
x
)
=
tan
x
Even or Odd Trigonometric Functions
Determine if the following functions are even, odd, or neither:
f
(
x
)
=
sin
x
+
cos
x
f\left(x\right)=\sin x+\cos x
f
(
x
)
=
sin
x
+
cos
x
and
g
(
x
)
=
sin
x
sec
x
g\left(x\right)=\dfrac{\sin x}{\sec x}
g
(
x
)
=
sec
x
sin
x
Q
:
\bf{Q:}
Q
:
Find all trigonometric function values of the angle
4
π
3
\dfrac{4\pi}{3}
3
4
π
Trigonometric Functions
Determine the number of solutions the equation
3
cos
x
+
sin
2
x
=
0
\sqrt{3}\cos x+\sin2x=0
3
cos
x
+
sin
2
x
=
0
has in the interval
[
0
,
2
π
]
\left[0,2\pi\right]
[
0
,
2
π
]
.
Practice: Trig Equation
Q
:
\bf{Q:}
Q
:
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Q
:
\bf{Q:}
Q
:
Given
cos
θ
=
12
13
\cos\theta=\dfrac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\dfrac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact value of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Practice: Special Angles
Q
:
\bf{Q:}
Q
:
Evaluate
tan
7
π
6
\tan\dfrac{7\pi}{6}
tan
6
7
π
and
sin
15
π
4
\sin\dfrac{15\pi}{4}
sin
4
15
π
.
Trigonometric Functions
Calculate the value of the expression
tan
−
1
(
tan
(
4
π
3
)
)
\tan^{-1}\left(\tan\left(\frac{4\pi}{3}\right)\right)
tan
−
1
(
tan
(
3
4
π
)
)
Practice: Trig Equation
Practice: Trig Equation
Solve the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
.
Practice: Trig Ratios
Practice: Trig Ratios
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Trigonometric Functions
Determine the number of solutions the equation
3
cos
(
x
)
+
sin
(
2
x
)
=
0
\sqrt{3}\cos\left(x\right)+\sin\left(2x\right)=0
3
cos
(
x
)
+
sin
(
2
x
)
=
0
has in the interval
[
0
,
2
π
]
\left[0,2\pi\right]
[
0
,
2
π
]
.
Practice: Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
.
Find the number of solutions to the equation
cos
2
x
=
cos
x
−
1
\cos2x=\cos x-1
cos
2
x
=
cos
x
−
1
in the interval [0,2𝜋].
Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify
1
−
cos
2
x
+
sin
2
x
cos
x
sin
x
1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}
1
−
cos
2
x
+
s
i
n
x
s
i
n
2
x
c
o
s
x
.
Trigonometric Functions
How many solutions does the equation
2
sin
2
x
=
3
sin
x
−
1
2\sin^2x=3\sin x-1
2
sin
2
x
=
3
sin
x
−
1
have on the interval
x
∈
[
0
,
2
π
]
x\in\left[0,2\pi\right]
x
∈
[
0
,
2
π
]
?
Trigonometric Functions
Given
cos
θ
=
12
13
\cos\ \theta=\frac{12}{13}
cos
θ
=
13
12
and
3
π
2
<
θ
<
2
π
\frac{3\pi}{2}<\theta<2\pi
2
3
π
<
θ
<
2
π
, find the exact values of
13
sin
2
θ
13\sin2\theta
13
sin
2
θ
.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
b. Simplify
2
log
5
10
+
log
5
2
−
log
5
8
2\log_5 10+\log_5 2-\log_5 8
2
lo
g
5
10
+
lo
g
5
2
−
lo
g
5
8
Trigonometric Functions: Trig Identities
Which of the following equations (if any) is true?
Trigonometric Functions
The body temperature
T
(
t
)
T(t)
T
(
t
)
of an individual over a day is a rhythmic process. It reaches a maximum of 38.0 C at 6:00 am and a minimum of 36.2 C at 6:00 pm. Choose the trigonometric function that best describes
T
(
t
)
T(t)
T
(
t
)
, where
t
t
t
is given in hours with
t
=
0
t = 0
t
=
0
taken at midnight (one minute after 11:59 pm, also written as 12:00 am).