High School Math 3 Unit 5a: Trigonometric Representation
Approximate Time Frame: 4-5 weeks
Connections to Previous Learning:
In Math 2 students learn trigonometric ratios and how to solve problems involving right triangles. They have also spent a significant amount of time in Math 1 and Math 2 interpreting, analyzing, and building new functions. They can interpret key features from a function or model, graph functions, find the average rate of change, and recognize transformations or shifts in a function or model.
Focus of this Unit:
Students will continue working on graphing and interpreting functions. They now focus on trigonometric functions: graphing, comparing representations, finding key features, and transformations or shifts. Students will extend their knowledge of trigonometric ratios to the unit circle, thus developing radian measure, and to the Pythagorean Identity, sin 2 (θ) + cos 2 (θ) = 1.
Connections to Subsequent Learning:
In the next unit students will focus on modeling and solving trigonometric functions. They will use their knowledge of trigonometric functions’ structure, key features, and graphs to work problems.
Grade 8, High School, Function Progressions document, p.17:
Trigonometric Functions
Right triangle trigonometry is concerned with ratios of sides of right triangles, allowing functions of angle measure to be defined in terms of these ratios. This limits the angles considered to those between 0 0 and 90 o . Circular trigonometry extends the domains of the trigonometric functions within the real numbers.
Traditionally, trigonometry includes six functions (sine, cosine, tangent, cotangent, secant, cosecant). Because the second three may be expressed as reciprocals of the first three, this progression discusses only the first three.
Extend the domain of trigonometric functions using the unit circle: In circular trigonometry, angles are usually measured in radians rather than degrees. Radian measure is defined so that in the unit circle the measure of an angle is equal to the length of the intercepted arc.F-TF.1 An angle of measure 1 radian turns out to be approximately57.3
0 . A full revolution, which corresponds to an angle of 360 0 , has measure equal to the circumference of the unit circle, or 2𝜋 . A quarter turn or
90 0 , measures 𝜋 /2 radians.
In geometry, students learn, by similarity, that the radian measure of an angle can be defined as the quotient of arc length to radius.
G-C.5
As a quotient of two lengths; therefore, radian measure is “dimensionless.” That is why the “unit” is often omitted when measuring angles in radians.
In calculus, the benefits of radian measure become plentiful, leading, for example, to simple formulas for derivatives and integrals of trigonometric functions. Before calculus, there are two key benefits of using radians rather than degrees:
• arc length is simply rθ, and
• sin 0 ≈ 0 for small θ.
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High School Math 3 Unit 5a: Trigonometric Representation
In right triangle trigonometry, angles must be between 0 0 and 90 0 . Circular trigonometry allows angles that describe any amount of rotation, including rotation greater than 360 0 .
F-TF.2
Consistent with conventions for measuring angles, counterclockwise rotation is associated with angles of positive measure and clockwise rotation is associated with angles of negative measure. The topic is the origin on the circle of radius 1 centered at the origin (called the unit circle), in the coordinate plane.
With the help of a picture, students mark the intended angle, 0 in radians, measured counterclockwise from the positive ray of the x-axis; identify the coordinates and x and y; draw a reference triangle; and then use right triangle trigonometry. In particular, 𝑠𝑖𝑛𝜃 = 𝑦 / 1 = 𝑦, cos 𝜃 = 𝑥 / 1 = 𝑥
And, 𝑡𝑎𝑛𝜃 = 𝑦/𝑥 . (Note the simplicity afforded by using a circle of radius 1. This way, students can compute values of any of the trigonometric functions, except that in a circular trigonometry being careful to note the signs of x and y. In the figure as drawn in the second quadrant, for example, is negative and is positive, which implies that sin 𝜃 is positive and cos 𝜃 and tan 𝜃 are both negative.
Grade 8, High School, Function Progressions document, p.20
Prove and apply trigonometric identities: For the cases illustrated by the diagram • (in which the terminal side of angle θ does not lie on • an axis) and the definitions of sin θ and cos θ, students can reason that, in any quadrant, the lengths of the legs of the reference triangle are 𝑥 and 𝑦 . It then follows from the Pythagorean Theorem that 𝑥 equation can be written
𝑠𝑖𝑛𝜃
2
(𝜃) + 𝑐𝑜𝑠
2 𝑥 2 + 𝑦
(𝜃) = 1 .
2 = 1 . Because 𝑐𝑜𝑠𝜃 and 𝑦 = 𝑠𝑖𝑛𝜃
2
+ 𝑦
2
= 1 . Because 𝑎
2
= 𝑎
2
for any real number a, this
. , the equation can be written as
When the terminal side of angle θ does lie on an axis, then one of x or y is 0 and the other is 1 or -1 and the equation still holds. This argument proves what is known as the Pythagorean identity F-TF.8
because it essentially a restatement of the
Pythagorean Theorem or 𝑡𝑎𝑛 𝜃 and the quadrant of the angle for a right triangle of hypotenuse 1.
With this identity and the value of one of the trigonometric function for a given angle, students can fins the values of the other functions for that angle, students can find the values of the other function for that angle, as long as they know the quadrant in which the angle lies. For example, if sin(𝜃) = 0.6
and 𝜃 lies in the second quadrant, then 𝑐𝑜𝑠 2 (𝜃) = 1 −
0.6
2
= 0.64
, so 𝑐𝑜𝑠
2 (𝜃) = ±√0.64 = ±0.8
. Because cosine is negative in the second quadrant, it follows that cos (𝜃) = 0.8
, and therefore tan(𝜃) =
= −0.75
.
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High School Math 3 Unit 5a: Trigonometric Representation
Extend the domain of trigonometric functions using the unit circle.

F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

F.TF.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.
Prove and apply trigonometric identities

F.TF.8 Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

A.SSE.2 Use the structure of an expression to identify ways to rewrite it.
Interpret functions that arise in applications in terms of the context.

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
★

F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Analyze functions using different representations

F.IF.7e graph exponential and logarithmic function, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Build new functions from existing functions.

F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:

Explicit vocabulary instruction with regard to the components and types of functions.

Explicit instruction with regard to visual models of trigonometric functions and their connection to algebraic representations.
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High School Math 3 Unit 5a: Trigonometric Representation

The unit circle allows all real numbers to work in trigonometric functions.

Trigonometric identities can be proven and used to solve problems with specified context.

Key features in graphs and tables shed light on relationships between two quantities.

Trigonometric functions can be represented by a table, graph, verbal description or equation, and each representation can be transferred to another representation

Specific transformations occur to trigonometric functions based on a value k and its manipulation to the function.

How do you use/read a unit circle (using radians)?

How does the Pythagorean theorem and the unit circle relate to the identity sin 2 (θ) + cos 2 (θ) = 1?

What do the key features of a trigonometric function represent?

What are the different ways you can represent a trigonometric function?

What transformations can occur to a trigonometric function/graph and how do you know which one is which?
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
*3. Construct viable arguments and critique the reasoning of others. Students will create informal arguments for the formula for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
4. Model with mathematics.
5. Use appropriate tools strategically.
*6. Attend to precision. Students will use the definitions and properties of trigonometric values to rewrite expressions and to prove trigonometric identities.
*7. Look for and make use of structure. Students will see how the structure of the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers
*8. Look for and express regularity in repeated reasoning. Students will understand how trigonometric values in quadrant one are related to trigonometric values in other quadrants.
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High School Math 3 Unit 5a: Trigonometric Representation

Define a function and the different parts in expressions, equations, and functions.

Define trigonometric ratios and solve problems involving right triangles.
Some students may be ready to:

Solve complex trigonometric functions.

Graph complex trigonometric functions.

Prove a variety of trigonometric identities.
Extend the domain of trigonometric functions using the unit circle

F.TF.3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6, and use the unit circle to express the values of sine, cosine, and tangent for –x, +x, and 2 –x in terms of their values for x, where x is any real number.

F.TF.4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions

F.TF.6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

F.TF.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
★
Prove and apply trigonometric identities
 F.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
:
 sin 2 (θ) + cos 2 (θ) = 1
:

Rewrite trigonometric expressions/equations to solve trigonometric identities.

Find the radian measure of an angle

Complete a unit circle and use it to determine trigonometric values of special angles.

Translate among representations of trigonometric functions including tables, graphs, equations, and real-life situations.

Graph a trigonometric function.

Calculate or interpret the rate of change of a function.
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Radian
Unit Circle
Amplitude
Frequency
Period
Midline
High School Math 3 Unit 5a: Trigonometric Representation
Arc
Trigonometric Ratio
Right Triangle
Sine, Cosine, Tangent
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