AP Precalculus Unit 3

3.1 - Periodic Phenomena

• Periodic Relationship - occurs between two variables when the output values demonstrates a repeating pattern over successive equal-length intervals. (Cyclical)

• Period of a Periodic Function - the length of the x - values that it takes for the function to complete one cycle.

3.2 - Sine, Cosine, and Tangent

• Standard Position - an angle is in standard position when its vertex is at the origin and one ray lies of the positive x - axis

• Terminal Ray - the second ray of an angle in standard position

Sine, Cosine, and Tangent

• Sin$\theta$ -

  • The ratio of the vertical displacement of P from the x - axis to the distance between the origin and point P.

  • $\sin\theta=\frac{y}{r}$

• Cos$\theta$ -

  • The ratio of the horizontal displacement of P from the y - axis to the distance between the origin and point P.

  • $\cos\theta=\frac{x}{r}$

• Tan$\theta$ -

  • The slope of the terminal ray - the ratio of the vertical displacement to the horizontal displacement of P

  • $\tan\theta=\frac{y}{x}$ or $\tan\theta=\frac{\sin\theta}{\cos\theta}$

where point P is where the terminal ray intersects the circle.

• Radian Angle Measure

  • Positive angles - counterclockwise direction

  • Negative angles - clockwise direction

• Measuring Angles

  • arc length / radius radians

  • Arc length can be found by using circumference

• Unit Circle

  • Radius is 1

  • Sine and cosine values of an angle are simplified

    • sin$\theta$ = y

    • cos$\theta$ = x

3.3 - Sine and Cosine Function Values

The coordinates of a point where a terminal ray intersects a circle will be $\left(r\cos\theta,r\sin\theta\right)$

• Finding Sine and Cosine Function Values

  • when working with the unit circle, pythagorean theorem can be used to find the x/y-coordinate or a point when already knowing the other coordinate

3.4 - Sine and Cosine Function graphs

• Properties of the graph for $f\left(\theta\right)=\sin\theta$

  

  • Midline - halfway between the max and min values

    • y = 0

  • Amplitude - distance from the midline to the max or min

    • a = 1

  • Period

    • P = 2$\pi$

  • Frequency - the reciprocal of the period

    • $\frac{1}{2\pi}$

• Properties of the graph for $g\left(\theta\right)=\cos\theta$

  

  • Midline - halfway between the max and min values

    • y = 0

  • Amplitude - distance from the midline to the max or min

    • a = 1

  • Period

    • P = 2$\pi$

  • Frequency - the reciprocal of the period

    • $\frac{1}{2\pi}$

3.5 - Sinusoidal Functions

• any function that involves additive and multiplicative transformations of $f\left(\theta\right)=\sin\theta$

• both sine and cosine functions are sinusoidal functions because $g\left(\theta\right)=\cos\theta=\sin\left(\theta+\frac{\pi}{2}\right)$

3.6 - Sinusoidal Function Transformations

using either function $f\left(\theta\right)=a\sin\left(b\theta\right)+d$ or $k\left(\theta\right)=a\cos\left(b\theta\right)+d$ the graphs will have the following transformations:

• Vertical Dilation by a factor of $a$ (amplitude of the graph)

• Vertical Translation by $d$ units (midline of the graph)

• Horizontal Dilation by a factor of $\left\vert\frac{1}{b}\right\vert$ (period is given by$P=\left\vert\frac{2\pi}{b}\right\vert$)

Phase Shift of a Sinusoidal Function

• horizontal translation of a sinusoidal function by $-c$ units

Parent Functions

• $f\left(\theta\right)=\cos\theta$

  • cosine is an even function, with reflective symmetry over the y - axis

  • cosine is a transformation of sine to the left

• $f\left(\theta\right)=\sin\theta$

  • sine is an odd function, with rotational symmetry about the origin.

3.7 - Sinusoidal Function Context and Data Modeling

literally just interpreting properties of a function to make construct a sinusoidal model

3.8 - The Tangent Function

$f\left(\theta\right)=\tan\theta$ , gives the slope of the terminal ray

• $\tan\left(\theta\right)=\frac{y}{x}$ as long as $\cos\theta\ne0$

• at $\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$ there is a vertical asymptote

Properties of the Tangent Function

• $f\left(\theta\right)=a\tan\left(b\left(\theta+c\right)\right)+d$

• vertical dilation by a factor of $a$

• Period $=\left\vert\frac{\pi}{b}\right\vert$

• Phase shift of $-c$ units

• Vertical Translation by $d$ units

3.9 - Inverse Trigonometric Functions

• Sine Function

  • Trig Function

    • $f\left(x\right)=\sin\left(x\right)$

    • Domain: $\left(-\infty,\infty\right)$

    • Range: $[-1,1\rbrack$

  • Inverse Trig Function

    • $f\left(x\right)=\sin^{-1}\left(x\right)$

    • Domain: $\left\lbrack-1,1\rbrack\right.$

    • Range: $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

      • the range must be restricted or else the inverse is not a function

  • Graphs:

• Cosine Function

  • Trig Function

    • $f\left(x\right)=\cos\left(x\right)$

    • Domain: $\left(-\infty,\infty\right)$

    • Range: $\left\lbrack-1,1\rbrack\right.$

  • Inverse Trig Function

    • $f\left(x\right)=\cos^{-1}\left(x\right)$

    • Domain: $\left\lbrack-1,1\rbrack\right.$

    • Range: $\left(0,\pi\right)$

      • the range must be restricted or else the inverse is not a function

  • Graphs:

• Tangent Function

  • Trig Function

    • $f\left(x\right)=\tan\left(x\right)$

    • Domain: $\left(-\infty,\infty\right)$ when $x\ne\frac{\pi}{2}+k\pi$ where $k$ is an integer

    • Range: $\left(-\infty,\infty\right)$

  • Inverse Trig Function

    • $f\left(x\right)=\tan^{-1}\left(x\right)$

    • Domain: $\left(-\infty,\infty\right)$

    • Range: $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

• Writing Inverse Functions

  • Switch the domain and the range

    • Sometimes you must restrict the domain

3.10 - Trigonometric Equations and Inequalities

Writing the General Solutions to a Trigonometric Equation

• $\sin\left(x\right)=\frac12$

  • $x=\frac{\pi}{6}+2\pi k$ , where $k$ is an integer

  • $x=\frac{5\pi}{6}+2\pi k$, where $k$ is an integer

• $\cos\left(x\right)=\frac12$

  • $x=\frac{\pi}{3}+2\pi k$ , where $k$ is an integer

  • $x=\frac{5\pi}{3}+2\pi k$ , where $k$ is an integer

• $\tan\left(x\right)=1$

  • $x=\frac{\pi}{4}+\pi k$ , where $k$ is an integer

Solving Trigonometric Equations

1. Isolate the trigonometric functions on one side of the equation

2. Find the corresponding angle measures on the unit circle that satisfy the given equation

3. Consider any domain restrictions in the problem

4. Write the solution and/or general solution

$\left(\sin x\right)^{n}=\sin^{n}x$

Solving Trigonometric Inequalities

1. Set f(x) = 0 and solve for x

2. Create a sign chart with the solution from step 1, include any domain restrictions

3. Test a value in each of the intervals

4. Label the remaining intervals as positive or negative

5. Interpret the sign chart to answer the inequality

3.11 - Secant, Cosecant, and Cotangent Functions

• Reciprocal Functions

  • Cosecant

    • $\csc x=\frac{1}{\sin x}$ , where $\sin x\ne0$

    • $y=\csc x$

      • vertical asymptote at $x=\pi k$ where $k$ is an integer

  • Secant

    • $\sec x=\frac{1}{\cos x}$ , where $\cos x\ne0$

    • $y=\sec x$

      • vertical asymptote at $x=\frac{\pi}{2}+\pi k$ where $k$ is an integer

  • Cotangent

    • $\cot x=\frac{1}{\tan x}$ , where $\tan x\ne0$

    • $\cot x=\frac{\cos x}{\sin x}$ , where $\sin x\ne0$

    • $y=\cot x$

      • vertical asymptote at $x=\pi k$ where $k$ is an integer

3.12 - Equivalent Representations of Trigonometric Functions

• The Pythagorean Identity and Equivalent Forms

  • $\sin^2\theta+\cos^2\theta=1$

  • $1+\tan^2\theta=\sec^2\theta$

  • $1+\cot^2\theta=\csc^2\theta$

• Trigonometric Identities

  • $\sin x=\frac{1}{\csc x}$

  • $\csc x=\frac{1}{\sin x}$

  • $\cos x=\frac{1}{\sec x}$

  • $\sec x=\frac{1}{\cos x}$

  • $\tan x=\frac{1}{\cot x}$

  • $\tan x=\frac{\sin x}{\cos x}$

  • $\cot x=\frac{1}{\tan x}$

  • $\cot x=\frac{\cos x}{\sin x}$

• Sum and Difference Identities

  • $\sin\left(a\pm b\right)=\sin a\cos b\pm\cos a\sin b$

  • $\cos\left(a\pm b\right)=\cos a\cos b-+\sin a\sin b$

    • (-+ means - or +, vs + or -)

• Double Angle Identities

  • $\sin\left(2\theta\right)=2\sin\theta\cos\theta$

  • $\cos\left(2\theta\right)=\cos^2\theta-\sin^2\theta,2\cos^2\theta-1,1-2\sin^2\theta$

3.13 - Trigonometry and Polar Coordinates

*WHEN USING POLAR GRAPHS MAKES SURE CALCULATOR IS ON POLAR GRAPH*

• Polar Coordinates - $\left(r,\theta\right)$

  • $\left\vert r\right\vert$ - radius of the circle

  • $\theta$ - angle in standard position whos terminal angle includes the point

• Converting from Polar to Rectangular Coordinates

  • $x=r\cos\theta$ and $y=rsin\theta$

• Converting from Rectangular to Polar Coordinates

  • $r^2=x^2+y^2$ and $\tan\theta=\frac{y}{x}$

• Complex Numbers

  • rectangular coordinates $\left(a,b\right)$ → $a+bi$

  • polar coordinates $\left(r,\theta\right)$ → $\left(r\cos\theta\right)+i\left(rsin\theta\right)$

• Converting Rectangular Complex Numbers to Polar Form

  • $r=\sqrt{x^2+y^2}$

  • $\theta=\tan^{-1}\left(\frac{y}{x}\right)$

• Converting Polar Complex Numbers to Rectangular Form

  • $x=r\cos\theta$

  • $y=rsin\theta$

3.14 - Polar Function Graphs

Circles

• $r=a\cos\theta$

  • $a$ positive - opens right

  • $a$ negative - opens left

  • Cycle: $[0,\pi\rbrack$

• $r=asin\theta$

  • $a$ positive - opens up

  • $a$ negative - opens down

Roses

• $r=a\cos\left(n\theta\right)$

  • starts on the polar axis

• $r=a\sin\left(n\theta\right)$

  • $n$ is odd - $n$ = # of petals

    • cycle: $[0,\pi\rbrack$

  • $n$ is even - 2$n$ = # of petals

    • cycle: $[0,2\pi\rbrack$

Limaçon

• $r=a\pm b\cos\left(\theta\right)$ or $r=a\pm b\sin\left(\theta\right)$

  • $a=b$ - Cardioid

    • distance from origin will be a + b

    • height will be = a

  • a>b , 1<\frac{a}{b}<2 Dimpled Cardioid (one loop limaçon)

    • a + b is distance from origin

    • point at origin will be shifted by a - b

    • height will be = a

  • a<b - Inner Loop Limaçon

    • b - a will be inner loop

    • same rules as the cardioid

3.15 - Rates of Change in Polar Function

Distance from the Pole

Average Rate of Change

• the rate of change for $r=f\left(\theta\right)$ is given by $\frac{f\left(b\right)-f\left(a\right)}{b-a}$

• to find a point using the rate of change use the following equation $r-$_ $=$ roc ($\theta-$ _ )