6.1 Sinusoidal Graphs

Periodic Function:

A function f is periodic if its values repeat at regular intervals.

Graphically, this means that if the graph of f is shifted horizontally by p units, the new graph is identical to the

original.

In Function Notation, periodic means that, for all x in the domain of f,

𝑓(𝑥 + 𝑝) = 𝑓(𝑥) The smallest positive constant p for which this relationship holds for all values of x is called the period of f.

The midline of a periodic function is the horizontal line midway between the function’s maximum and minimum

values.

The amplitude is the vertical distance between the function’s maximum (or minimum) value and the midline.

Graphs of the Sine and Cosine Functions (Note: Sinusoids are considered to be the general form of the sine and

cosine function)

𝑡 0 𝜋

4

𝜋

2

3𝜋

4 𝜋

5𝜋

4

3𝜋

2

7𝜋

4 2𝜋

sin 𝑡

cos 𝑡

Period: _________

Midline: ________

Amplitude: ______

Period: _________

Midline: ________

Amplitude: ______

Amplitude

𝑦 = 𝑎 sin 𝑡,

then |𝑎| is the amplitude of the graph.

Ex. 𝑦 = sin 𝑡

𝑦 = 2 sin 𝑡

𝑦 = − sin 𝑡

Midline

The graph of 𝑦 = 𝑓(𝑡) + 𝑑 is the graph of 𝑦 = 𝑓(𝑡) shifted vertically by d units.

Note: If 𝑦 = sin 𝑡 + 𝑑 and 𝑦 = cos 𝑡 + 𝑑, we have midlines 𝑦 = 𝑑.

Ex. 𝑦 = cos 𝑡 + 2

Note:

Midline of both graphs is 𝑦 = 0.

Amplitude is 1.

Note: If a is negative, the graph is reflected across the t-axis

Shifted Sine and Cosine Curves Given an equation in the form khtBAtf )(sin)( or khtBAtf )(cos)(

amplitude = |𝐴|

period, B

P2

midline 𝑦 = 𝑘

h is the horizontal shift of the function

An appropriate interval on which to graph one complete period is[ℎ, ℎ + (2𝜋𝐵⁄ )]

Note: horizontal shift means: the curves shifted horizontally by an amount |ℎ|. They are shifted to the right if h >

0 or to the left if h < 0.

Period

Ex. 𝑦 = sin(2𝑡) Ex. 𝑦 = sin(𝑡2⁄ )

Horizontal Shift

Ex. 𝑦 = 2sin(𝑡 + 𝜋2⁄ ) Ex. 𝑦 = −3sin(2𝑡 − 𝜋

2⁄ )

For the equation below, determine the amplitude, midline, period, and horizontal shift (indicated the direction).

𝑦 = −5sin(𝜋𝑡 + 2) − 4

For the graphs below, determine the amplitude, midline, period, and horizontal shift (indicated the direction), then write an

equation for the graph.

5. 7.

17.

23. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meters above the ground. The six o'clock

position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The

function ( )h t gives your height in meters above the ground t minutes after the wheel begins to turn.

a. Find the amplitude, midline, and period of h t

b. Find an equation for the height function h t

c. How high are you off the ground after 5 minutes?

6.2 Graphs of the Other Trigonometric Functions

On a unit circle, we know that

𝑥 = and 𝑦 =

Find the slope of the line 𝑂𝑃 ,

Graph of the Tangent Function (𝒚 = 𝒕𝒂𝒏𝜽)

Period =

Range:

Domain:

Period =

Range:

Domain:

P(x, y)

1

θ y

x 0

Note: The functions 𝑦 = 𝐴 tan(𝐵𝑥) and 𝑦 = 𝐴 cot(𝐵𝑥), where 𝐵 > 0, have period 𝜋

𝐵.

Graph of 𝑦 = csc 𝑥 Graph of 𝑦 = sec 𝑥

Find the period and horizontal shift of each of the following functions,

5. 2tan 4 32f x x

7. 2sec 14

h x x

and then graph the function

21. If tan 1.5x , find tan x

23. If sec 2x , find sec x

Period =

Range:

Domain:

Period =

Range:

Domain:

Note: The functions 𝑦 = 𝐴 csc(𝐵𝑥) and 𝑦 = 𝐴 sec(𝐵𝑥), where 𝐵 > 0, have period 2𝜋

𝐵.

6.3 Inverse Trigonometric Functions

Recall:

If f is one-to-one function with domain A and range B, then its inverse f-1

is the function with domain B and range

A defined by

𝑓−1(𝑥) = 𝑦 ⇔ 𝑓(𝑦) = 𝑥 Therefore,

f-1

is the rule that reverse the action of f. (From output back to input)

Note: 𝑓−1 reads f inverse or inverse of f, and xf

xf11

If a function has an inverse, it is said to be invertible.

The Inverse Sine Functions:

The inverse Sine function is the function 𝑠𝑖𝑛−1 with domain [-1, 1] and range

2,

2

defined by

sin−1 𝑥 = 𝑦 ⇔ sin 𝑦 = 𝑥

The inverse Sine function is also called the arcsine function, denoted by xarcsin .

Evaluate the following expressions

1. 1 2sin

2

2. 1 1sin

2

3.

1sin 1

4.

6sinsin 1

5.

2

1sinsin 1 6. 5sinsin 1

Domain:

2,

2

Domain: [-1, 1]

Range: [-1, 1] Range:

2,

2

The Inverse Cosine Functions: The inverse Cosine function is the function 𝑐𝑜𝑠−1 with domain [-1, 1] and range ,0 defined by

cos−1 𝑥 = 𝑦 ⇔ cos 𝑦 = 𝑥

The inverse Cosine function is also called the arccosine function, denoted by xarccos .

Evaluate the following expressions

1. 1 1cos

2

2. 1 2cos

2

3. 0cos 1

4.

3

4coscos 1

5. 3.0coscos 1

6. 2coscos 1

Domain: ,0 Domain: [-1, 1]

Range: [-1, 1] Range: ,0

The Inverse Tangent Functions:

The inverse Tangent function is the function 𝑡𝑎𝑛−1 with domain , and range

2,

2

defined by

tan−1 𝑥 = 𝑦 ⇔ tan 𝑦 = 𝑥

The inverse Tangent function is also called the arctangent function, denoted by xarctan .

Evaluate the following expressions

1. 1tan 1 2. 1tan 3 3.

3tantan 1

Evaluate the following expressions

4.

4cossin 1

5.

3

4cossin 1

6.

7

3sincos 1 7. 4tancos 1

8.

5

4cossin 1

Find a simplified expression for each of the following

9.

5cossin 1 x

, for 55 x 10. x3tansin 1

Domain:

2,

2

Domain: ,

Range: , Range:

2,

2

xx

xx

1

1

tantan

tantan

for

for

The cancellation equations are

−𝜋

2< 𝑥 <

𝜋

2

x

6.4 Solving Trig Equations Find the solutions of the equation: a) 0 ≤ 𝑥 ≤ 𝜋; b) 0 ≤ 𝑥 < 2𝜋; c) All Solutions

cos 𝑥 = −1

2

Solving Trig Equations: To solve a trigonometric equation:

1) Use the rules of algebra to isolate the trigonometric functions on one side of the equal sign.

2) Make a substitution for the inside of the sine or cosine, if it is other then x or θ.

3) Use our knowledge of the values of the trigonometric functions to solve for the variable:

Use factoring techniques

Use trigonometric identities

4) Use the inverse trig functions to find one solution

5) Use symmetries to find a second solution on one cycle (when a second exists)

6) Find additional solutions if needed by adding full periods

Note: To get all other solutions:

add any integer multiple of 2π to solutions related to Sine and Cosine, and

add any integer multiple of π to solutions related to Tangent.

7) Undo the substitution

Find all solutions on the interval 0 2

1. 2sin 2 2. 2cos 1

3. sin 1 4. 3tan3

Find all solutions

9. 2cos 2 11. 2sin 1 12. 1tan

Find all solutions on the interval 0 2

13. 2sin 3 1

17. 2cos 2 1

21. cos 14

27. 58.0 sin x

Note: In solving a trig. Equation for aθ, in which the argument is not θ, like 2sin 3 1 , you must write

down all the solutions first, then solve for θ .

6.5 Modeling with Trigonometric Equations Solving right triangles In each of the following triangles, solve for the unknown side and angles (in degree).

2.

Modeling with sinusoidal functions Problems that involve quantities that oscillate can often be modeled by a sine or cosine function and once we create a

suitable model for the problem we can use the equation and function values to answer the question.

Find a possible formula for the trigonometric function whose values are in the following tables. 6.

x 0 1 2 3 4 5 6

y 1 -3 -7 -3 1 -3 -7

8. Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature for

the day is 92 degrees and the low temperature of 78 degrees occurs at 4 AM. Assuming t is the number of hours since

midnight, find an equation for the temperature, D, in terms of t.

B

7

3

A

c

10. A population of elk oscillates 150 above and below an average of 720 during the year, hitting the lowest value in

January (t = 0).

a. Find an equation for the population, P, in terms of the months since January, t.

b. What if the lowest value of the elk population occurred in March instead?

12. Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of

84 degrees occurs at 6 PM and the average temperature for the day is 70 degrees. Find the temperature, to the nearest

degree, at 7 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.