Warm-Up

Draw the Unit Circle on a sheet of paper and label ALL the parts.

DO NOT LOOK AT YOUR NOTES

So far, in our study of trigonometric functions we have:

•defined all of them•learned how to evaluate them •used them on the unit circle

So logically, the next step would be to study the graphs of the functions.

Unit 2: Graphs and Inverses of Trigonometric Functions

LG 2-1 Graphing Trig Functions (Quiz 9/6

LG 2-2 Circular Functions (Quiz 9/10)

LG 2-3 Evaluating Inverse Trig Functions (Quiz 9/12)

LG 2-4 Graphing Inverse Trig Functions (Quiz 9/14)TEST 9/17

What’s Your Temperature?

Scientists are continually monitoring the average temperatures across the globe to determine if Earth is experiencing Climate Change (Global Warming!).

One statistic scientists use to describe the climate of an area is average temperature. The average temperature of a region is the mean of its average high and low temperatures.

What’s Your Temperature? A function that repeats itself in regular

intervals, or periods, is called periodic.

a. If you were to continue the temperature graph, what would you consider its interval, or period, to be?

b. Choose either the high or low average temperatures and sketch the graph for three intervals, or periods.

Periodic Functions• A periodic function is a function whose values

repeat at regular intervals. – Sine and Cosine are examples of periodic functions

• The part of the graph from any point to the point where the graph starts repeating itself is called a cycle.

• The period is the difference between the horizontal coordinates corresponding to one cycle.– Sine and Cosine functions complete a cycle every 360°.

So the period of these functions is 360°.

Periodicity is Common in Nature• Day/night cycle (rotation of

earth) • Ocean Tides • Pendulums and other swinging

movements • Ocean waves • Birth/marriage/death cycle • Menstrual cycle • Eating and sleeping cycle • Musical rhythm • Linguistic rhythm

• Dribbling, juggling • Calendars • Fashion cycles, for example, skirt

lengths or necktie widths • Economic and political cycles, for

example, boom and bust economic periods, right-wing and left-wing political tendencies

Whenever you have to draw a graph of an unfamiliar function, you must do it by point-wise plotting, or

calculate and plot enough points to detect a pattern. Then you connect the points with a smooth curve or line.

Objective: Discover by point-wise plotting what the graphs of the six trig functions look like.

Homework: Finish the characteristics table and do the discovery task

LG 2-1 Graphing Trigonometric Functions

Graphing Calculator Introduction

• Put your calculator in degree mode.

• Graph y = sin Θ. Trace along the graph. What do you observe?

• Repeat for y = cos Θ but this time put your calculator on radian mode.

Graphing Sine and Cosine Functions• The graph of the sine and cosine functions are

made by evaluating each function at the special angles on the unit circle.

• The input of the function is the angle (in degrees OR radians) measure on the unit circle.

• The output is the value of the function for that angle.

• We can “unwrap” these values from the unit circle and put them on the coordinate plane.

Exploration: Parent Sinusoids

Sinusoid – a graph of a sine or cosine function

“sinus” coming from the same origin as “sine,” and “– oid” being a suffix meaning “like.”

The Graph of Sinex (angles) y

(evaluate for sine)

0o

90o

180o

270o

360o

The Graph of Cosinex (angles) y

(evaluate for cos)

0o

90o

180o

270o

360o

Important Words• Sinusoidal axis - the horizontal line halfway between the

local maximum and local minimum: y = 0 for parent function• Convex – bulging side of the wave• Concave – hollowed out side of the wave

– Concave up– Concave down

• Point of inflection - point on a curve at which the sign of the curvature (the concavity) changes.

The Graph of Tangentx (angles) y

(evaluate for tan)

0o

90o

180o

270o

360o

Co-Trig Functions (Reciprocal Functions)

• Each of the co-functions relate to the original graph.

• Plot the “important points” for the sine function on the cosecant graph and then sketch the sine curve LIGHTLY in pencil (borrow one if you need to!)

The Graph of Cosecantx (angles) y

(evaluate for csc)

0o

90o

180o

270o

360o

The Graph of Secantx (angles) y

(evaluate for sec)

0o

90o

180o

270o

360o

The Graph of Cotangentx (angles) y

(evaluate for cot)

0o

90o

180o

270o

360o

Discontinuous Functions

• The graphs of tan, cot, sec, and csc functions are discontinuous where the function value would involve division by zero. What happens to the graph when a function is discontinuous?

Trig Function Characteristics

• As always, we need to talk about domain, range, max and min, etc.

• You will fill out the table for each of the 6 trig functions (and finish for HW).

Period• The period of a trig function is how long it

takes to complete one cycle. • What is the period of sine and cosine? • What about cosecant and secant?• Tangent and cotangent?• The period of the functions tangent and

cotangent is only 180° instead of 360°, like the four trigonometric functions.

Domain and Range

• When we think about the domain and range, we have to make sure we are considering the entire function and not just the part on the unit circle.

Maximum and Minimum

• Local max & min• Absolute max & min

Intercepts

• x-intercepts• y-intercepts

Points of Inflection

• point on a curve at which the sign of the curvature (the concavity) changes.

• Will all graphs have points of inflection?

Intervals of Increase and Decrease

• Positive slope? • Negative slope?

Classwork…

• Fill out the characteristics table for each of the 6 trig functions

Function Period Domain Range

y = tan Θ 180° Θ = all real #’s of degrees except Θ = 90° + 180°n, where n is an integer

y = all real #’s

y = cot Θ 180° Θ = all real #’s of degrees except Θ = 180°n, where n is an integer

y = all real #’s

y = sec Θ 360° Θ = all real #’s of degrees except Θ = 90° + 180°n, where n is an integer

y = csc Θ 360° Θ = all real #’s of degrees except Θ = 180°n, where n is an integer

1 or 1y y

1 or 1y y

What’s Your Temperature?

Sine and Cosine functions can be used to model average temperatures for cities.

Based on what you learned about these graphs, why do you think these functions

are more appropriate than a cubic function? Or an exponential function?