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

Trigonometric Functions

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Trigonometric (Trig) Functions

SOH CAH TOA

Using a right triangle, Trig Functions can be defined by using the hypotenuse, opposite side, and adjacent sides.
sin⁡θ=opphypcsc⁡θ=hypoppcos⁡θ=adjhypsec⁡θ=hypadjtan⁡θ=oppadjcot⁡θ=adjopp\begin{array}{ccc} \sin\theta=\frac{opp}{hyp} & &\csc\theta=\frac{hyp}{opp}\\ \\ \cos\theta=\frac{adj}{hyp} & & \sec\theta=\frac{hyp}{adj}\\ \\ \tan\theta=\frac{opp}{adj} & &\cot\theta=\frac{adj}{opp} \end{array}


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CAST Rule (All Students Take Calculus)

We can determine the sign (positive or negative) for our trig functions based on which quadrant the angle measure falls in.


• Quadrant 1: All trig ratios are positive• Quadrant 2: Only sin ratio is positive• Quadrant 3: Only tan ratio is positive• Quadrant 4: Only cos ratio is positive\begin{array}{l} •\ \text{Quadrant 1: All trig ratios are positive}\\ •\ \text{Quadrant 2: Only sin ratio is positive}\\ •\ \text{Quadrant 3: Only tan ratio is positive}\\ •\ \text{Quadrant 4: Only cos ratio is positive} \end{array}

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Trig Graphs


Special Triangles

So called 30-60-90 Triangles and 45-45-90 Triangles help us find exact values of trig functions.


  1. Draw out the angle given
  2. Identify the relative acute angle (RAA)--angle made with the x-axis
  3. Find the trig ratio of the RAA using special triangles
  4. Make the trig ratio positive or negative based on the CAST rule
Example: tan⁡(π3)=\tan\left(\frac{\pi}{3}\right)=


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Example: Trig Values

Evaluate the following

a) tan⁡7π6\displaystyle \tan\frac{7\pi}{6}
  1. The angle 7π6\frac{7\pi}{6} can be written as π+π6\pi+\frac{\pi}{6}. So, when we draw out the angle, it’s in quadrant 3.
  2. The relative acute angle (RRA) is the angle it makes with the 𝑥-axis, which will be π6\frac{\pi}{6}.
  3. Finding the trig. ratio of the relative acute angle, we get tan⁡(π6)=13\tan\left(\frac{\pi}{6}\right)=\frac{1}{\sqrt{3}}
  4. Since tan⁡\tan is positive in the 3rd quadrant (use CAST rule), our answer is tan⁡(7π6)=13\tan\left(\frac{7\pi}{6}\right)=\frac{1}{\sqrt{3}}


b) sin⁡15π4\displaystyle \sin\frac{15\pi}{4}
  1. The angle 15π4\frac{15\pi}{4} can be written as 3π+3π43\pi+\frac{3\pi}{4}. So, when we draw out the angle, it's in quadrant 4.
  2. The relative acute angle (RRA) is the angle it makes with the 𝑥-axis, which will be π4\frac{\pi}{4}.
  3. Finding the trig. ratio of the relative acute angle, we get sin⁡(π4)=12\sin\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}
  4. Since sin⁡\sin is negative in the 4th quadrant (use CAST rule), out answer is sin⁡(15π4)=−12\sin\left(\frac{15\pi}{4}\right)=-\frac{1}{\sqrt{2}}
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Trig Identities

Basic Identities

  • tan⁡θ=sin⁡θcos⁡θ\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}
  • sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1
  • tan⁡2θ+1=sec⁡2θ\tan^2\theta+1=\sec^2\theta

Double Angles

  • sin⁡(2θ)=2sin⁡θ cos⁡θ\sin\left(2\theta\right)=2\sin\theta\ \cos\theta
  • cos⁡(2θ)=cos⁡2θ−sin⁡2θ  or  2cos⁡2θ−1  or  1−2sin⁡2θ\cos\left(2\theta\right)=\cos^2\theta-\sin^2\theta\ \ \text{or}\ \ 2\cos^2\theta-1\ \ \text{or}\ \ 1-2\sin^2\theta

Angle Sum & Difference

  • sin⁡(x±y)=sin⁡xcos⁡y±cos⁡xsin⁡y\sin\left(x\pm y\right)=\sin x\cos y\pm\cos x\sin y
  • cos⁡(x±y)=cos⁡xcos⁡y∓sin⁡xsin⁡y\cos\left(x\pm y\right)=\cos x\cos y\mp\sin x\sin y

Even and Odd Properties

  • Odd function: sin⁡(−θ)=−sin⁡θ\sin\left(-\theta\right)=-\sin\theta
  • Even function: cos⁡(−θ)=cos⁡θ\cos\left(-\theta\right)=\cos\theta
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.

Solve the equation 2sin⁡2x=3sin⁡x−12\sin^2x=3\sin x-1 on the interval [0,2π]\left[0,2\pi\right].


Extra Practice
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: Trigonometric Modeling
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?
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).