High School
SAT
ACT
University
MCAT
LSAT
DAT

Practice: Trigonometric Modeling

Related Topics

Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Inverse Trigonometric Functions

4 Activities

Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Trigonometric Functions

5 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Q:\bf{Q:} The temperature of the body oscillates throughout the day and can be modeled by the function: T(t)=37 + 0.4cos⁡(π12 t−π3)T\left(t\right)=37\ +\ 0.4\cos\left(\dfrac{\pi}{12} \ t-\dfrac{\pi}{3}\right) where TT is the temperature in degrees Celsius, and tt 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?
c) What is the phase shift of this function? What does it tell you about the time of day when the maximum temperature is reached?
d) What is the physical meaning of the inverse function of TT^{ }?
e) Find the inverse function of TT^{ }

Note: This question is for practice purposes only, it is not based on real scientific data.
More Inverse Trigonometric Functions Questions:
Inverse Trigonometric Functions
Compute the exact value of
arccos⁡(cos⁡(3π4))\arccos \left ( \cos \left ( \frac{3 \pi}{4} \right ) \right )
Find the exact value of arcsin⁡(sin⁡(−17π6))\arcsin(\sin(-\frac{17\pi}{6}))
Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Trig & Inverse Trig Values
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
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)=1arcsin⁡xf\left(x\right)=\frac{1}{\arcsin x}.
Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Trig & Inverse Trig Values
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Inverse Trigonometric Functions
Evaluate tan⁡(tan⁡−1π4)+arcsin⁡(sin⁡5π4)\tan\left(\tan^{-1}\frac{\pi}{4}\right)+\arcsin\left(\sin\frac{5\pi}{4}\right).
Inverse Trigonometric Functions
Find the exact value of cos⁡[tan⁡−12+tan⁡−13]\cos[\tan^{-1}2+\tan^{-1}3].
Inverse Trig with Product
Find the exact value of sin⁡[2cos⁡−1(15)]\sin\bigg[2\cos^{-1}\bigg(\dfrac{1}{5}\bigg)\bigg].
Inverse Trigonometric Functions: Restrict the Domain
Find the exact value of arccos⁡(cos⁡23π3)\arccos\bigg(\cos\dfrac{23\pi}{3}\bigg) .
Inverse Trigonometric Functions
Simplify 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⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Inverse Trigonometric Functions
Without the using a calculator, compute the exact value of
arctan⁡(tan⁡(13π11))\arctan \left ( \tan \left ( \frac{13\pi}{11} \right ) \right )
*The π\pi 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⁡(3π4))\arccos \left ( \cos \left ( \frac{3 \pi}{4} \right ) \right )
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for x:
a. 9x−2=(127)2x+19^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
Inverse Trigonometric Functions
Find the exact value of sin⁡(2cos⁡−1(x))\sin\left(2\cos^{-1}\left(x\right)\right).
Inverse Trigonometric Functions
Find the value of sin (arccos (− √3/2 )).
Inverse Trigonometric Functions
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Inverse Trigonometric Functions
Simplify the following expression:
sec⁡2(arctan⁡(1/x2))\sec^2(\arctan(1/x^2))
Inverse Trigonometric Functions
Simplify
sin⁡(2arcsin⁡x)\sin(2\arcsin x)
Inverse Trigonometric Functions
Find the exact value of tan⁡[arcsin⁡(−3/5)]\tan[\arcsin(-3/5)]
Inverse Trigonometric Functions
Find the exact value of sin⁡(2cos⁡−1x)\sin(2\cos^{-1}x)
Inverse Trigonometric Functions
Find the domain and range of f(x)=1arcsin⁡xf(x) = \frac{1}{\arcsin x}
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for x:
a. 9x−2=(127)2x+19^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
Simplify the following expression: sin⁡[−tan⁡−1(x2−1)]\sin[-\tan^{-1}(\sqrt{x^2-1})]
Inverse Trigonometric Functions: Restrict the Domain
Find the exact value of arccos⁡(cos⁡23π3)\arccos\bigg(\cos\dfrac{23\pi}{3}\bigg) .
Inverse Trigonometric Functions
Find the exact value of cos⁡[tan⁡−12+tan⁡−13]\cos[\tan^{-1}2+\tan^{-1}3].
Inverse Trig with Product
Find the exact value of sin⁡[2cos⁡−1(15)]\sin\bigg[2\cos^{-1}\bigg(\dfrac{1}{5}\bigg)\bigg].
Inverse Trigonometric Functions
Simplify the expression cos⁡(tan⁡−1(x2))\cos\left(\tan^{-1}\left(\frac{x}{2}\right)\right)
Inverse Trigonometric Functions
Simplify the following expression;
sec⁡−1(2)=\sec^{-1}(2)=
Inverse Trigonometric Functions
Calculate the value of the expression sin⁡−1(sin⁡(5π3))\sin^{-1}\left(\sin\left(\frac{5\pi}{3}\right)\right)
Find the exact value of arcsin⁡(sin⁡(−17π6))\arcsin(\sin(-\frac{17\pi}{6}))
Inverse Trigonometric Functions
Find sin⁡(arctan⁡(x))\sin\left(\arctan\left(x\right)\right).
Practice: Trig & Inverse Trig Values (level 2)
Practice: Trig & Inverse Trig Values
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Practice: Trig & Inverse Trig Values (level 3)
Practice: Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Practice: Domain & Range with Inverse Trig
Practice: Domain & Range with Inverse Trig
Find the domain and range of f(x)=1arcsin⁡xf\left(x\right)=\frac{1}{\arcsin x}.
Find the value of sin⁡[arccos⁡(−32)]\sin\left[\arccos\left(-\frac{\sqrt{3}}{2}\right)\right] .
Inverse Trigonometric Functions
Find sin (arctan 𝑥).
Trig & Inverse Trig Values
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
Trig & Inverse Trig Values
Find the exact value of sin⁡(2cos⁡−1x)\sin\left(2\cos^{-1}x\right).
Domain & Range with Inverse Trig
Find the domain and range of f(x)=1arcsin⁡xf\left(x\right)=\frac{1}{\arcsin x}.
Inverse trigonometric functions
Evaluate the expression arcsin⁡(−12)×cos⁡(7π6)+2ln⁡(3e)\arcsin\left(-\frac{1}{\sqrt{2}}\right)\times\cos\left(\frac{7\pi}{6}\right)+2\ln\left(3\sqrt{e}\right)
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Inverse Trigonometric Functions
Evaluate tan⁡(tan⁡−1π4)+arcsin⁡(sin⁡5π4)\tan\left(\tan^{-1}\frac{\pi}{4}\right)+\arcsin\left(\sin\frac{5\pi}{4}\right).
Practice: Trig & Inverse Trig Values (level 2)
Find the exact value of tan⁡[arcsin⁡(−35)]\tan\left[\arcsin\left(-\frac{3}{5}\right)\right].
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 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Evaluate 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Evaluate 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_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⁡(3e2)+e2ln⁡3\ln\left(3e^2\right)+e^{2\ln3}
Trigonometric Functions: Trig Identities
Which of the following equations (if any) is true?
Trigonometric Functions
Given cos⁡ θ=1213\cos\ \theta=\frac{12}{13} and 3π2<θ<2π\frac{3\pi}{2}<\theta<2\pi , find the exact values of 13sin⁡2θ13\sin2\theta.
Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify 1−cos⁡2x+sin⁡2x cos⁡xsin⁡x1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}.
Trigonometric Functions
How many solutions does the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 have on the interval x∈[0,2π]x\in\left[0,2\pi\right]?
Practice: Trig Equation
Q:\bf{Q:} How many solutions does the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 have on the interval x∈[0,2π]x\in\left[0,2\pi\right]?
Q:\bf{Q:} Given cos⁡θ=1213\cos\theta=\dfrac{12}{13} and 3π2<θ<2π\dfrac{3\pi}{2}<\theta<2\pi , find the exact value of 13sin⁡2θ13\sin2\theta.
Practice: Special Angles
Q:\bf{Q:} Evaluate tan⁡7π6\tan\dfrac{7\pi}{6} and sin⁡15π4\sin\dfrac{15\pi}{4}.
Q:\bf{Q:}Find all trigonometric function values of the angle 4π3\dfrac{4\pi}{3}
Q:\bf{Q:} Find the domain of the following function: f(x)=tan⁡ xf\left(x\right)=\tan\ x
Trigonometric Functions
Determine the number of solutions the equation 3cos⁡x+sin⁡2x=0\sqrt{3}\cos x+\sin2x=0 has in the interval [0,2π]\left[0,2\pi\right].
Even or Odd Trigonometric Functions
Determine if the following functions are even, odd, or neither: f(x)=sin⁡x+cos⁡xf\left(x\right)=\sin x+\cos x and g(x)=sin⁡xsec⁡xg\left(x\right)=\dfrac{\sin x}{\sec x}
Exponential and Trigonometric Functions
Q:\bf{Q:} Find the domain of . f(x)=sin⁡(3x)e4x−1−tan⁡(2x)f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
Logarithmic and Trigonometric Functions
Solve for x in the equation ln⁡(4−4sin⁡2x)=0\ln\left(4-4\sin^2x\right)=0
Trigonometric Functions
Solve the following equation: 2sin⁡x−23cos⁡x=3tan⁡x−32\sin x-2\sqrt{3}\cos x=\sqrt{3}\tan x-3
Trigonometric Functions
Given 3cos⁡x+2sin⁡2x=33\cos x+2\sin^2x=3 , solve for xx in the domain (0,−3π2]\left(0,-\dfrac{3\pi}{2}\right].
Practice: Transforming a Trig Function
Graph the function: y=3sin⁡(2x+π2)+1\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+1
Trigonometric Functions
The solutions of the equation tan⁡2x=sin⁡2x+cos⁡2x\tan^2x=\sin^2x+\cos^2x in the interval [0,π]\left[0,\pi\right] are
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Practice: Transforming a Trig Function
Graph the function: y=3sin⁡(2x+π2)+1\displaystyle y=3\sin\left(2x+\frac{\pi}{2}\right)+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) measure the temperature of water at time t (in minutes). Which of the following could be an equation for 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⁡(2x)=cos⁡(x)−1\cos\left(2x\right)=\cos\left(x\right)-1 in the interval [0,π]\left[0,\pi\right].
Greatest Integer Function
Let f(x)=[[sin⁡x−2]]f\left(x\right)=\left[\left[\sin x-2\right]\right] be the greatest integer of sin⁡x−2\sin x-2 on the open interval (0, π)\left(0,\ \pi\right).
The range of f(x)f(x) is
Trigonometric Functions
Find all solutions to sin⁡(θ)=sin⁡(2θ)\sin(\theta)=\sin(2\theta) on the interval [0,2π)[0,2\pi)
Trigonometric Functions
Simplify 1−cos⁡2x+sin⁡2xcos⁡xsin⁡x1-\cos2x + \frac{\sin 2x \cos x}{\sin x}
Additional Functions Practice Problems
Additional Practice Problems--Functions
1. Solve the following equations for x:
a. 9x−2=(127)2x+19^{x-2}=\left(\frac{1}{27}\right)^{2x+1}
Practice: Trig Equation
Practice: Trig Equation
Solve the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 on the interval x∈[0,2π]x\in\left[0,2\pi\right].
Practice: Trig Ratios
Practice: Trig Ratios
Given cos⁡ θ=1213\cos\ \theta=\frac{12}{13} and 3π2<θ<2π\frac{3\pi}{2}<\theta<2\pi , find the exact values of 13sin⁡2θ13\sin2\theta.
Practice: Trig Equation
Practice: Trig Equation
How many solutions does the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 have on the interval x∈[0,2π]x\in\left[0,2\pi\right]?
Logarithmic and Trigonometric Functions
Solve for x in the equation ln⁡(4−4sin⁡2x)=0\ln\left(4-4\sin^2x\right)=0
Trigonometric Functions
Given 3cos⁡x+2sin⁡2x=33\cos x+2\sin^2x=3 , solve for xx in the domain (0,−3π2]\left(0,-\dfrac{3\pi}{2}\right].
Trigonometric Functions
Solve the following equation: 2sin⁡x−23cos⁡x=3tan⁡x−32\sin x-2\sqrt{3}\cos x=\sqrt{3}\tan x-3
Exponential and Trigonometric Functions
Q:\bf{Q:} Find the domain of . f(x)=sin⁡(3x)e4x−1−tan⁡(2x)f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
Q:\bf{Q:} Find the domain of the following function: f(x)=tan⁡ xf\left(x\right)=\tan\ x
Even or Odd Trigonometric Functions
Determine if the following functions are even, odd, or neither: f(x)=sin⁡x+cos⁡xf\left(x\right)=\sin x+\cos x and g(x)=sin⁡xsec⁡xg\left(x\right)=\dfrac{\sin x}{\sec x}
Q:\bf{Q:}Find all trigonometric function values of the angle 4π3\dfrac{4\pi}{3}
Trigonometric Functions
Determine the number of solutions the equation 3cos⁡x+sin⁡2x=0\sqrt{3}\cos x+\sin2x=0 has in the interval [0,2π]\left[0,2\pi\right].
Practice: Trig Equation
Q:\bf{Q:} How many solutions does the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 have on the interval x∈[0,2π]x\in\left[0,2\pi\right]?
Q:\bf{Q:} Given cos⁡θ=1213\cos\theta=\dfrac{12}{13} and 3π2<θ<2π\dfrac{3\pi}{2}<\theta<2\pi , find the exact value of 13sin⁡2θ13\sin2\theta.
Practice: Special Angles
Q:\bf{Q:} Evaluate tan⁡7π6\tan\dfrac{7\pi}{6} and sin⁡15π4\sin\dfrac{15\pi}{4}.
Trigonometric Functions
Calculate the value of the expression tan⁡−1(tan⁡(4π3))\tan^{-1}\left(\tan\left(\frac{4\pi}{3}\right)\right)
Practice: Trig Equation
Practice: Trig Equation
Solve the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 on the interval x∈[0,2π]x\in\left[0,2\pi\right].
Practice: Trig Ratios
Practice: Trig Ratios
Given cos⁡ θ=1213\cos\ \theta=\frac{12}{13} and 3π2<θ<2π\frac{3\pi}{2}<\theta<2\pi , find the exact values of 13sin⁡2θ13\sin2\theta.
Trigonometric Functions
Determine the number of solutions the equation 3cos⁡(x)+sin⁡(2x)=0\sqrt{3}\cos\left(x\right)+\sin\left(2x\right)=0 has in the interval [0,2π]\left[0,2\pi\right].
Practice: Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify 1−cos⁡2x+sin⁡2x cos⁡xsin⁡x1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}.
Find the number of solutions to the equation cos⁡2x=cos⁡x−1\cos2x=\cos x-1 in the interval [0,2𝜋].
Simplifying Trig Expressions
Practice: Simplifying Trig Expressions
Simplify 1−cos⁡2x+sin⁡2x cos⁡xsin⁡x1-\cos2x+\frac{\sin2x\ \cos x}{\sin x}.
Trigonometric Functions
How many solutions does the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 have on the interval x∈[0,2π]x\in\left[0,2\pi\right]?
Trigonometric Functions
Given cos⁡ θ=1213\cos\ \theta=\frac{12}{13} and 3π2<θ<2π\frac{3\pi}{2}<\theta<2\pi , find the exact values of 13sin⁡2θ13\sin2\theta.
Inverse Trigonometric Functions
Which of the following results in a value that is positive:
Additional Practice Problems--Functions Part 2
b. Simplify 2log⁡510+log⁡52−log⁡582\log_5 10+\log_5 2-\log_5 8
Trigonometric Functions: Trig Identities
Which of the following equations (if any) is true?
Trigonometric Functions
The body temperature 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) , where tt is given in hours with t=0t = 0 taken at midnight (one minute after 11:59 pm, also written as 12:00 am).