1. Trigonometric Functions
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Trigonometric Functions
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2. Information Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Information
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3. Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Periodic functions
A function with a pattern that repeats at regular intervals is called a periodic function.
The length of the interval over which a periodic function repeats is called the period of the function.
Mathematical practices
6) Attend to precision.
Students should use and understand the term “period” when applied to functions.
7) Look for and make use of structure.
Students should look for repeating patterns in functions and identify equivalent points in order to find the period.
What is the period of the function shown above?
period = 4
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4. Trigonometric functions of real numbers
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Trigonometric functions of real numbers
The trigonometric functions are common periodic functions.
For any real number θ, the trigonometric functions are defined:
sinθ = y/r
cosθ = x/r
tanθ = y/x
where x and y are the coordinates of any point P(x, y) on the terminal ray of an angle of θ radians when it is in standard position, and r is the distance from the origin to P. r θ
Teacher notes
This extends the definitions of trigonometric ratios to all real numbers.
See the presentation Unit Circle and Radians for more information.
In particular, if P is on the unit circle:
sinθ = y
cosθ = x
tanθ = y/x
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5. The graph of sinθ Boardworks High School Algebra II (Common Core)
Trigonometric Functions
The graph of sinθ
Teacher notes
Ask students to identify key features of the graph. Notice that the curve repeats itself every 2π radians or 360°, and show how this relates to the unit circle by dragging the point around the full circle more than once. Other patterns or identities that could be recognized are:
sinx = –sin(–x)
sinx = sin(x + 2π)
sinx = sin(π – x)
Mathematical practices
7) Look for and make use of structure.
8) Look for and express regularity in repeated reasoning.
Students should notice how the absolute value of sinθ follows the same pattern in each quadrant as the angle exceeds 2π and should express these patterns as identities.
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6. Sine parent function: f (x) = sinx Is sinx an even or odd function?
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Sine
parent function:
f (x) = sinx
domain:
(– ∞, ∞)
range:
[–1, 1]
asymptotes:
none
period: 2π
roots:
x = nπ
Teacher notes
Sine is odd. This means that sin–x = –sinx and the graph has rotational symmetry of order 2 around the origin.
Mathematical practices
5) Use appropriate tools strategically.
Students can use graphing calculators to investigate the properties of the function.
7) Look for and make use of structure.
Students should be able to relate “even” and “odd” to earlier analysis of functions.
maxima:
x = π⁄2 + 2nπ
minima:
x = –π⁄2 + 2nπ
for every integer n
Is sinx an even or odd function?
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7. The graph of cosθ Boardworks High School Algebra II (Common Core)
Trigonometric Functions
The graph of cosθ
Teacher notes
Ask students to identify key features of the graph. Notice that the curve repeats itself every 2π radians or 360°, and show how this relates to the unit circle by dragging the point around the full circle more than once. The cosine curve is symmetrical about the vertical axis. Other patterns or identities that could be recognized are:
cosx = cos(–x)
cosx = cos(x + 2π)
cosx = cos(2π –x)
Students should also be guided to look at the similarities with the sine curve and the connection: sinx = cos(x – π/2) or cosx = sin(x + π/2)
Mathematical practices
7) Look for and make use of structure.
8) Look for and express regularity in repeated reasoning.
Students should notice how the absolute value of cosθ follows the same pattern in each quadrant and as the angle exceeds 2π and should express these patterns as identities.
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8. Cosine parent function: f (x) = cosx Is cosx an even or odd function?
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Cosine
parent function:
f (x) = cosx
domain:
(– ∞, ∞)
range:
[–1, 1]
asymptotes:
none
period: 2π
roots:
x = π⁄2 + nπ
Teacher notes
Cosine is even. This means that cos–x = cosx and the graph is symmetrical about the y-axis.
Mathematical practices
5) Use appropriate tools strategically.
Students can use graphing calculators to investigate the properties of the function.
7) Look for and make use of structure.
Students should be able to relate “even” and “odd” to earlier analysis of functions.
maxima:
x = 2nπ
minima:
x = π + 2nπ
for every integer n
Is cosx an even or odd function?
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9. The graph of tanθ Boardworks High School Algebra II (Common Core)
Trigonometric Functions
The graph of tanθ
Teacher notes
Ask students to identify key features of the graph. Notice that the curve repeats itself every π radians or 180°. The dashed lines represent asymptotes, showing that tanθ is undefined at π /2 (90°) and nπ + π/2 for integer values of n.
Other patterns or identities that could be recognized are:
tanx = –tan(–x)
tanx = tan(x + π)
Mathematical practices
7) Look for and make use of structure.
8) Look for and express regularity in repeated reasoning.
Students should notice how the absolute value of tanθ follows the same pattern in each quadrant and should express these patterns as identities.
© Boardworks 2013

10. Tangent parent function: f (x) = tanx Is tanx an even or odd function?
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Tangent
parent function:
f (x) = tanx
domain:
(nπ – π⁄2 , nπ + π⁄2 )
range:
(– ∞, ∞)
period: π
roots:
x = nπ
horizontal asymptotes:
none
Teacher notes
Tan is odd. This means that tan–x = –tanx and the graph has rotational symmetry of order 2 around the origin.
Mathematical practices
5) Use appropriate tools strategically.
Students can use graphing calculators to investigate the properties of the function.
7) Look for and make use of structure.
Students should be able to relate “even” and “odd” to earlier analysis of functions.
vertical asymptotes:
x = π⁄2 + nπ
for every integer n
Is tanx an even or odd function?
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11. Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Amplitude and midline
A key feature of a periodic function is its amplitude.
The amplitude of a periodic function is half the difference between the minimum and maximum y values, (maxy – miny)/2.
amplitude
midline
Teacher notes
The amplitude is 2 and the equation of the midline is y = 3.
Mathematical practices
6) Attend to precision.
Students should use and understand the terms “amplitude” and “midline” when applied to periodic functions.
The midline of a periodic function is the line at the midpoint of the range. y = miny + (maxy – miny)/2
What is the amplitude of the graph shown?
What is the equation of its midline?
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12. Trigonometric functions
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Trigonometric functions
Teacher notes
For the claims that are false, ask students to correct them. For example, question 2 should be “sinx and cosx have an amplitude of 1,” question 3 should be “the midline of cosx is y = 0”, and question 4 should be “The period of tanx is half the period of sinx.”
Mathematical practices
3) Construct viable arguments and critique the reasoning of others.
Students should be able to explain why each statement is true or false, using diagrams if necessary.
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13. Transforming trigonometric graphs
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Transforming trigonometric graphs
Teacher notes
Use this activity to explore transformations of sine, cosine and tangent graphs. The input is x here, rather than θ, to be consistent with general function notation. This might be a good opportunity to talk about cycles.
Mathematical practices
7) Look for and make use of structure.
Students should look for patterns in how each transformation affects the functions.
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14. Transformations and key features
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Transformations and key features
Teacher notes
Emphasize that these relationships only apply to the parent trigonometric functions. If the midline is not at y = 0, then a transformation nf (x) will also affect the location of the midline, as well as the amplitude.
Mathematical practices
1) Make sense of problems and persevere in solving them.
Students should think about how the transformations affect a trigonometric graph and relate this to the terminology they have learnt for key features of periodic functions.
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15. Analyzing functions Graph f (x) = –3cos2x.
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Analyzing functions
Graph f (x) = –3cos2x.
1) Find the domain, range, period and amplitude.
2) Find the x-values where the minimum and maximum values occur in the interval [0, 2π].
3) State the zeros in the interval [0, 2π]. 1)
domain:
(– ∞, ∞)
range:
[– 3, 3]
period: π
amplitude: 3
Teacher notes
The graph shown has the window [–2π, 2π] for x and [–3, 3] for y.
Mathematical practices
5) Use appropriate tools strategically.
Students should use a graphing calculator to graph the function and calculate the values. 2)
minima:
x = 0, π, 2π
maxima:
x = π/2, 3π/2 3)
zeroes:
x = π/4, 3π/4, 5π/4, 7π/4
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16. General form Boardworks High School Algebra II (Common Core)
Trigonometric Functions
General form
Teacher notes
Ask students to figure out which expression gives each key feature by trying out different numbers and looking at the graphs produced. Figuring this out for themselves will help students remember the relationships.
Mathematical practices
3) Construct viable arguments and critique the reasoning of others.
Students should sketch a mathematical argument to link each variable with a key feature.
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17. Match the graph to the equation
Boardworks High School Algebra II (Common Core)
Trigonometric Functions
Match the graph to the equation
Mathematical practices
1) Make sense of problems and persevere in solving them.
6) Attend to precision.
7) Look for and make use of structure.
8) Look for and express regularity in repeated reasoning.
Students should read the graph carefully to find the midline, amplitude, period and maxima and minima, so that they can calculate the equation.
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