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T3.4 - Graphs of Trigonometric Functions

IB Math SL1 - Santowski

Lesson Objectives

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Understand the trig ratios of angles in the context of angles in standard position in a co-ordinate plane rather than triangles

Graph and analyze a sinusoidal function

Make the connection between angles in standard position and sinusoidal functions

Fast Five

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Evaluate sin(50°) illustrate with a diagramEvaluate sin(130°) illustrate with a diagram

Evaluate sin(230°) illustrate with a diagram??Evaluate sin(320°) illustrate with a diagram??Evaluate sin(770°0 illustrate with a diagram??

Evaluate sin(-50°) illustrate with a diagram??

Use your calculator and graph the function f(x) = sin(x) on the domain -720° < x < 720°

(A) Angles in Standard Position

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Angles in standard position are defined as angles drawn in the Cartesian plane where the initial arm of the angle is on the x axis, the vertex is on the origin and the terminal arm is somewhere in one of the four quadrants on the Cartesian plane

To form angles of various measure, the terminal arm is simply rotated through a given angle

(B) Terms Related to Standard Angles

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Quadrant I lies between 0° - 90°

Quadrant II lies between 90° - 180°

Quadrant III lies between 180° and 270°

Quadrant IV lies between 270° and 360°

(B) Terms Related to Standard Angles

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A principle angle is any angle between 0° and 360°

A coterminal angle is one which shares the same terminal arm and the same initial arm as a principle angle, but was formed by several rotations of the terminal arm, so that it winds up in the same position as the terminal arm of its principle angle. Draw an example

A negative angle is one which is formed from a rotation in a clockwise direction. Draw an example

a related acute angle is the angle between the x axis and the terminal arm and will always be between 0° and 90°. Draw an example

(B) Terms Related to Standard Angles

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(C) Angles in Standard Position – Interactive Applet

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Go to the link below and work through the ideas presented so far with respect to angles in standard position

Angles In Trigonometry from AnalyzeMath

(D) Examples

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ex 1. Draw a 225° angle and label the principle angle and the related acute angle and draw one coterminal angle.

ex 2. Determine and draw the next two consecutive positive coterminal angles and the first negative coterminal angle with 43°

ex 3. Draw a –225° and label the principle angle and the related acute angle and draw one coterminal angle

(E) Trig Ratios of Angles in Standard Position

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(E) Trig Ratios of Angles in Standard Position

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We can once again set up our angle in the Cartesian plane and now simply determine the sin, cos, and tan ratios of these angles as we had in our previous lessons:

We simply place a point on the terminal arm, determine its x,y coordinates and then drop a perpendicular from the point down to the x axis. So now we have our right triangle.

As such, we can now define the primary trig ratios as follows:

sine A = y/rcosine A = x/rtangent A = y/x

(F) Examples

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we will move the point A(3,4) through the four quadrants and determine the sine, cosine and tangent ratios in each of the four quadrants:

Quadrant I - P(3,4) sin A = 4/5, cos A = 3/5, tan A = 4/3

Quadrant II - P(-3,4) sin A = 4/5, cos A = -3/5, tan A = -4/3

Quadrant III - P(-3,-4) sin A = -4/5, cos A = -3/5, tan A = 4/3

Quadrant IV - P(3,-4) sin A = -4/5, cos A = 3/5, tan A = -4/3

(I) Examples

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Ex 1. The terminal arm of an angle goes through the point (-3,5).

(i) draw a diagram showing the angle, (ii) determine the angle’s three primary trig ratios, (iii) illustrate the related acute angle(iv) determine the angle that corresponds to each

of the primary ratios. Interpret.

Ex 2. The cosine ratio of an angle is –4/7. Draw the angle in standard position and determine the other trig ratios for the angle. What is the measure of the angle? Include a diagram

(J) Internet Links

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Topics in trigonometry:   Measurement of angles from The Math Page

Angles In Trigonometry from AnalyzeMath

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(A) Graph of f(x) = sin(x)

We can use our knowledge of angles on Cartesian plane and our knowledge of the trig ratios of special angles to create a list of points to generate a graph of f(x) = sin(x)

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(A) Graph of f(x) = sin(x)We have the following points from the first quadrant

that we can graph: (0,0), (30,0.5), (45,0.71), (60,0.87) and (90,1)

We have the following second quadrant points that we can graph: (120,0.87), (135,0.71), (150,0.5), and (180,0)

We have the following third quadrant points: (210,-0.50), (225,-0.71), (240,-0.87) and (270,-1)

Finally we have the 4th quadrant points: (300,-0.87), (315,-.71), (330,-0.5) and (360,0)

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(A) Graph of f(x) = sin(x)Now we need to consider the co-terminal angles as well

to see what happens beyond our one rotation around the 4 quadrants

For example, consider that sin(390) is the sine ratio of the first positive coterminal angle with 390-360 = 30 degrees

So, sin(390) = sin(30) = 0.5

So we can extend our list of points to include the following:

(390,0.5), (405,0.71), (420,0.87) and (450,1)(480,0.87), (495,0.71), (510,0.5), and (540,0)(570,-0.50), (585,-0.71), (600,-0.87) and (630,-1)(660,-0.87), (675,-.71), (690,-0.5) and (720,0)

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(A) Graph of f(x) = sin(x)Now we need to consider the negative angles as

well to see what happens by rotating “backwards”

For example, consider that sin(-30) is the sine ratio of the first negative coterminal angle with 360-30 = 330 degrees

So, sin(-30) = sin(330) = -0.5

So we can extend our list of points to include the following:

(-30,-0.5), (-45,-0.71), (-60,-0.87) and (-90,-1)(-120,-0.87), (-135,-0.71), (-150,-0.5), and (180,0)(-210,0.50), (-225,0.71), (-240,0.87) and (-270,1)(-300,0.87), (-315,.71), (-330,0.5) and (-360,0)

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(A) Graph of f(x) = sin(x)

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(A) Features of f(x) = sin(x) The graph is periodic (meaning that it repeats

itself) Domain: Range: Period: length of one cycle, how long does the pattern

take before it repeats itself . x-intercepts: amplitude: max height above equilibrium position - how

high or low do you get y-intercept: max. points: min. points:

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(A) Features of f(x) = sin(x) The graph is periodic (meaning that it repeats

itself) Domain: x E R Range: [-1,1] Period: length of one cycle, how long does the pattern

take before it repeats itself 360 or 2 π rad. x-intercepts: every 180 x = 180n where n E I or πn

where n E I. amplitude: max height above equilibrium position - how

high or low do you get => 1 unit y-intercept: (0,0) max. points: 90+ 360n (or 2π + 2 π n) min. points: 270+ 360n or -90 + 360n or -π/2 + 2 π

n

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(B) Graph of f(x) = cos(x)We can repeat the same process of

listing points and plotting them to see the graph of f(x) = cos(x)

Our first quadrant points include:(0,1), (30,0.87), (45,0.71), (60,0.5) and

(90,0)And then we could list all the other

points as well, or simply turn to graphing technology and generate the graph:

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(B) Graph of f(x) = cos(x)

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(B) Features of f(x) = cos(x) The graph is periodic Domain: Range: Period: length of one cycle, how long does the pattern

take before it repeats itself . x-intercepts: amplitude: max height above equilibrium position - how

high or low do you get y-intercept: max. points: min. points:

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(B) Features of f(x) = cos(x) The graph is periodic Domain: x E R Range: [-1,1] Period: length of one cycle, how long does the pattern

take before it repeats itself 360 or 2 π rad. x-intercepts: every 180 starting at 90, x = 90 +

180n where n E I (or π/2 + π n where n E I) amplitude: max height above equilibrium position - how

high or low do you get => 1 unit y-intercept: (0,1) max. points: 0 + 360n ( 2 π n) min. points: 180 + 360n or -180 + 360n (or π + 2

π n)

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(C) Graph of f(x) = tan(x)Likewise, for the tangent function, we

list points and plot them:(0,0), (30,0.58), (45,1), (60,1.7),

(90,undefined)(120,-1.7), (135,-1), (150,-0.58), (180,0)(210, 0.58), (225,1), (240,1.7),

(270,undefined)(300,-1.7), (315,-1), (330,-0.58), (360,0)

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(C) Graph of f(x) = tan(x)

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(C) Features of f(x) = tan(x) The graph is periodic Domain: Asymptotes: Range: Period: length of one cycle, how long does the pattern

take before it repeats itself x-intercepts: amplitude: max height above equilibrium position - how

high or low do you get y-intercept: max. points: min. points:

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(C) Features of f(x) = tan(x) The graph is periodic Domain: x E R where x cannot equal 90, 270, 450, or

basically 90 + 180n where n E I Asymptotes: every 180 starting at 90 Range: x E R Period: length of one cycle, how long does the pattern

take before it repeats itself = 180 or π rad. x-intercepts: x = 0, 180, 360, or basically 180n where

n E I or x = πn amplitude: max height above equilibrium position - how

high or low do you get => none as it stretches on infinitely y-intercept: (0,0) max. points: none min. points: none

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(D) Internet LinksUnit Circle and Trigonometric Functions

sin(x), cos(x), tan(x) from AnalyzeMath

Relating the unit circle with the graphs of sin, cos, tan from Maths Online

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(E) HomeworkNelson text, Section 5.2, p420, Q1-9eol, 11-15Section 5.3, p433, Q1-3,

13,14,20,21,22,24,25

Haese Text, Ex 13D.1, Q1-4,6Ex 13D.2, Q1f,5hjEx 13G, Q1chi,4Ex 13K.2, Q1,2