Unit Circle & Graphing
Essential Question: How can we model periodic (repeating) and fluctuating (oscillating) data?
Angles
1) Degree and Radian Measurements and Conversion
- Explanation:
- Notes | Annotated
- Videos:
- Khan: Introduction to Radians
- Khan: Radians & degrees
- Khan: Degrees to radians
- Khan: Radians to degrees
- Website that animates the number of radians in a circle
- Explanation and extra examples from purplemath (at the bottom of the page)
- Practice Problems:
- Worksheet #2
- Khan Exercise: Degrees to Radians
2) Coterminal and Reference Angles
- Practice Problems:
Trig Operations
3) Trig on a Coordinate Plane
4) Completing the Unit Circle:
- Explanation:
- Completed Unit Circle:
- Notes
- Free app in iTunes
- Khan video: Introduction to the unit circle
- Khan article: Trig unit circle review
- Completed Unit Circle:
- Practice:
5) Exact Trig Ratios on a Unit Circle:
- Explanation:
- Notes | Annotated Notes
- Khan article: Trig ratios of special triangles
- Khan video: Unit circle
- Khan article: Trig unit circle review
- Khan video: Trig values of special angles
- Desmos graph (physical distances of all six trig ratios)
- Practice Problems:
- Quizlet
- Khan Exercise: Trig values of special angles (only sine and cosine)
- Worksheet #1
- Worksheet #2
- Worksheet #3
6) Inverse Trig Ratios on a Unit Circle:
- Explanation:
- Notes | Annotated Notes
- Khan article: Inverse trigonometric functions review
- Khan video: Intro to arcsine
- Khan video: Intro to arctangent
- Khan video: Intro to arccosine
- Practice Problems:
- Khan exercise: Evaluate inverse trig functions (with a calculator)
- Worksheet #1
- Worksheet #2
- Worksheet #3
7) Order of Operations with Trig Ratios:
- Explanation:
- Practice Problems:
- Worksheet #3
Graphing Trig Functions
8) Graphing Sin and Cosine from the Unit Circle:
- Explanation:
- Khan Video: Graph of y=sin(x)
- Khan video: Intersection points of y=sin(x) and y=cos(x)
- Interactive graphs of the relationship between the unit circle and the graph of sin and cos curve:
- Large Coordinate Plane handout
- Modified directions from Illuminations: Graphs from the Unit Circle
- Practice Problems:
- Graphs from the Unit Circle WS
9) Transformations of sin and cos: Amplitude and Period
- Explanation:
- Desmos Activity: Polygraph: Sinusoids with Vertical Transformations (teacher use only)
- Blank Trig Coordinate Planes
- Interactive Websites:
- Graph sin with change in amplitude with Desmos
- Graph cos with change in amplitude with Desmos
- Graph sin with change in period (multiplying by a number) with Desmos
- Graph sin with change in period (dividing by a number) with Desmos
- Graph cos with change in period (multiplying by a number) with Desmos
- Graph cos with change in Period (dividing by a number) with Desmos
- NLVM (Interactive website using java by the Utah State University - NLVM): graph any trig function; also make your own graph with up to three sliding variables
- video of frequency of noise and water
- Practice Problems:
- Worksheet #1
- Worksheet #2
10) Transformation of sin and cos: Vertical Shift (shift the x-axis or "midline")
- Explanation
- Practice Problems:
- Khan Exercise: Graph sinusoidal functions (challenging)
- Worksheet #1
- Worksheet #2
- Worksheet #3
11) Transformation of sin and cos: Horizontal Shift (shift the y-axis or Phase Shift)
- Explanation:
- Interactive graph of sin (desmos)
- Interactive graph of cosine (desmos)
- Practice Problems
- Worksheet #2
- Worksheet #3
- Khan Exercise: Graph sinusoidal functions: phase shift (challenging)
12) Identify Parts of a Trig Function from an Equation (and Graphing Mixed Review)
- Explanation:
- Identify Parts of a Trig Function: Notes | Annotated
- Graphing Mixed Review: Notes | Annotated
- Khan video: Amplitude & period of sinusoidal functions from equation
- Sound Waves
- Practice Problems:
- Desmos Polygraph: Sinusoids (Teacher Use Only)
- Khan exercise: Midline of sinusoidal functions from equation
- Khan exercise: Amplitude of sinusoidal functions from equation
- Khan exercise: Period of sinusoidal functions from equation (give answers in terms of radians)
- Identify Parts of a Trig Function WS #1
- Identify Parts of a Trig Function WS #2
- Graphing Sin and Cosine by Transformations (Mixed Review):
13) Writing Sine and Cosine Functions
- Explanation
- Notes | Annotated
- vocabulary review
- Khan article: Midline, amplitude, and period review
- Khan Video: Features of sinusoidal functions
- Khan Video: Sinusoidal function from graph
- Khan Video: Trig word problem: modeling daily temperature
- Khan Video: Trig word problem: modeling annual temperature
- Khan Video: Trig word problem: length of day (phase shift)
- Khan video: Predator-prey cycles
- Practice Problems:
- Desmos Activity: Trigonometric Graphing: Introduction to Amplitude and Vertical Shift (Teacher use only)
- Desmos Activity: Graphing the Sine Function using Amplitude, Period, and Vertical Translation (Teacher use only)
- Desmos Activity: Marbleslides (Teacher use only)
- Desmos Activity: Burning Daylight (Teacher use only)
- Worksheet #1
- Worksheet #2
- Khan Exercise: Midline of sinusoidal functions from graph
- Khan Exercise: Amplitude of sinusoidal functions from graph
- Khan Exercise: Period of sinusoidal functions from graph
- Khan Exercise: Construct sinusoidal functions
- Khan Exercise: Modeling with sinusoidal functions
- Khan Exercise: Modeling with sinusoidal functions: phase shift
- Math Modeling with Trig Project Foxes and Rabbits
14) Graphing Secant & Cosecant Functions
- Explanation:
15) Graphing Tangent & Cotangent Functions
- Explanation
- Notes | Annotated
- Kahn Video: Graph of y=tan(x)
- Desmos: example of transforming tangent
- Desmos: example of transforming cotangent
- Practice Problems:
- Worksheet #1
- Worksheet #2
16) Unit Circle and Graphing Trig Functions Review
Common Core Standard: Functions: Trigonometric Functions
- Extend the domain of trigonometric functions using the unit circle:
- A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
- A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
- A.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any real number.
- Model periodic phenomena with trigonometric functions:
- B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Resources:
http://www.intmath.com/trigonometric-graphs/trigo-graph-intro.php