AC Precalculus

Unit 2 – Trigonometric Graphs

Lesson 4.1 (Blue): Graphs of Sine and Cosine Functions

Objective: In this lesson you will learn how to graph the basic sine and cosine functions. You will use the amplitude and the period to help sketch the graphs of the sine and cosine functions.

Periodic Function: A periodic function is a function f such that f ( x )=f (x+ p) for every real number x in the domain of f and for some positive real number p. The smallest value of p is the period of the function.

To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points, and minimum points.

As x increases from sin x cos x

0 to π2

π2 to π

π to 3π2

3π2 to 2π

Example 1: Sketch the graph of

y=2sin x on[−2π ,2π ].

Example 2: Sketch the graph of y=3cos x on the interval [−360 ° ,360 ° ] .

Amplitude of Sine and Cosine – The graph of y=a sin x and y=acos x, with a≠0, will have the same shape as the graph of y=sin x or y=cos x, respectively, except with range [−|a|,|a|]. The number |a| is called the amplitude.

Example 3: On the same set of axes, sketch the graph of each function.

a. y=12

cos x

b. y=−2cos x

Set your graphing window to: = −2π = 2π

π /2

Graph: f ( x )=sin x g ( x )=sin 2x h ( x )=sin 12x

How do the graphs compare?

Scaling: Horizontal Stretching

By dividing the interval __________ into four equal parts, we obtain the values for which sin bx or cosbx is -1, 0, 1. These will give minimum points, x-intercepts, and maximum points on the graph. Once these points are determined, the graph can be completed by joining the points with a smooth sinusidoidal curve.

Example 4: Analyze the graph of y=−2sin 3 x . Then sketch a graph of the equation.

Example 5: Analyze the graph of

y=3cos 12x . Then sketch a graph of the

equation.

AC Precalculus

Unit 2 – Trigonometric Graphs

Lesson 4.2 (Blue): Translating Graphs of Sine and Cosine Functions

Objective: In this lesson you will learn how to translate the graphs of the sine and cosine functions using phase shifts (horizontal translations) and vertical translations.

Horizontal TranslationsIn general, the graph of a function y=f (x−d) is translated horizontally when compared with the graph of y= f (x ). The translation is d units to the right if _________ and ________ units to the left if ___________. With circular functions, a horizontal translation is called a _______________________________. In the function y=f (x−d), the expression x−d is called the argument.

Example 1: Graph y=sin(x−π3 ) .Method 1 Method 2

Example 2: Graph y=3cos (x+ π4 ).

Example 3: Graph y=−2cos (3x+π ).

Example 4 : Analyze the graph of y=−3cos (2πx+4 π ) . Then sketch a graph of the equation.

Example 5: Analyze the graph of

y=−12

sin (πx+π ) . Then sketch a graph of the equation.

Vertical TranslationThe graph of a function of the form y=c+ f (x ) is shifted vertically as compared with the graph of y=f (x ). The function y=c+ f (x ) is called a vertical translation of y=f (x ).

Example 6: Analyze the graph of y=2cos x−5. Then sketch a graph of the equation.

Example 7: Analyze the graph of y=−1+2sin 4(x+ π4 ).

Graphing General Sine and Cosine FunctionsTo graph the general function y=c+a sin b ( x−d ) or y=c+acosb(x−d), where b > 0, follow these steps.

1. Find an interval whose length is one period (2πb ) by solving the compound inequality

0≤b(x−d)≤2 π 2. Divide the interval into four equal parts.3. Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum

points, minimum points, and points that intersect the line y=c (“middle” points of the wave).4. Plot the points found in Step 3, and join them with a sinusoidal curve.5. Draw the graph over additonal points, to the right and to the left, as needed.

Example 8: Use the steps above to identify the five critical values of the graph of y=−52

+cos (3 x−π2 ) .

AC Precalculus

Unit 2 – Trigonometric Graphs

Lesson 4.5 (GREEN): Graphs of Sine and Cosine Functions

Objective: In this lesson you will learn to use the sine and cosine functions to model real-life data.

Example 1: Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The table shows the depths (in feet) at various times during the morning. (Source: Nautical Software, Inc.)

a. Use a trigonometric function to model the data.b. Find the depths at 9 a.m. and 3 p.m.c. A boat needs at least 10 feet of water to moor at the dock. During what

times in the afternoon can it safely dock?

Example 2: The normal monthly temperatures in degrees Fahrenheit in Albany, NY for six months are given in the table.

a. Use a trigonometric function to model this data.b. Find the normal temperature in December.c. A painting company will accept exterior jobs only when the normal

temperature is 64 degrees or higher. During what months will this company accept exterior jobs?

x (month) y (temperature)

Jan 21Mar 34May 58Jul 72Sep 61Nov 40

AC Precalculus

Unit 2 – Trigonometric Graphs

Lesson 4.3 (BLUE): Graphs of the Other Circular Functions

Objective: In this lesson you will learn how to sketch the graphs of the secant, cosecant, tangent and cotangent functions.Graph f ( x )=sin x and g ( x )=cos x

y=csc x

y=sec x

Period: Period:

Domain: Domain:

Range: Range:

Vertical Asymptote: Vertical Asymptote:

Symmetry: Symmetry:

Example 1: Sketch the graph of y=2csc (x+ π4 ).

Example 2: Sketch the graph of y=sec 2x

Example 3: Sketch the graph of y=2csc( x+ π2 ).

Example 4: Sketch the graph of y=sec πx .

x −π2

-1.57 -1.5 −π4

0 π4

1.5 1.57 π2

tan x

y=tan x= sin xcos x

bx−c=−π2 bx−c=π

2

y=a tan(bx−c)

Example 5: Sketch the graph of each tangent function.

a. y=tan( x2 )

b. y=tan 2x

c. y=−3 tan 2x

The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of π. However, from the identity

y=cot x=cos xsin x . Asymptotes:

x 0 π4

π2

3π4

π

cot x

y=tan x= sin xcos x

bx−c=0 bx−c=π

y=acot(bx−c)

Example 6: Sketch the graph of each cotangent function.

a. y=cot x4

b. y=2cot x3

c. y=cot(2 x)

AC Precalculus

Unit 2 – Trigonometric Graphs

Lesson 4.7: Inverse Trigonometric Functions

Recall that for a function to have an inverse function, it must be one-to-one - that is, it must pass the Horizontal Line Test.

If you restrict the domain to the interval _____________________ the following properties hold.

1. On the interval ____________________, the function y = sin x is increasing.2. On the interval y= sin x takes on its full range of values, −1≤sin x ≤1.3. On the interval y = sin x is one-to-one.

So, on the restricted domain __________________________, y = sin x has a unique inverse function called the inverse sine function.It is denoted by ______________________ or _______________________.

Example 1: If possible, find the exact value.

a. arcsin (−12 ) b. sin−1 √3

2 c. sin−12

d. arcsin (−1) e. sin−1 12 f. sin−1√3

Example 2: Sketch the graph of y=arcsin x.

Other Inverse Trigonometric Functions

Example 3: Find the exact value.

a. arccos √22

b. cos−1(−1) c. arccos √32

d. arctan 1 e. tan−1 √33

Calculators and Inverse Trigonometric Functions

Example 4: Use a calculator to approximate the value (if possible).a. arctan (−8.45) b. sin−10.2447c. arccos(−0.349) d. sin−1(−1.1) e. arccos 2

Composition of Functions

Example 5: If possible, find the exact value.

a. tan ¿ b. arcsin (sin 5π3 ) c. cos ¿ d. tan ¿e. sin ¿

Right triangles can be used to find exact values of compositions of inverse functions.

Example 6: Find the exact value.

a. tan(arccos 23 ) b. cos¿¿

Example 7: Write each of the following as an algebraic expression in x.a. sin ¿

b. cot ¿

c. sec ¿

d. tan¿¿

AC Precalculus

Unit 2 – Trigonometric Graphs

Lesson 4.8: Applications and Models

Objective: In this lesson you will learn to solve problems involving right triangles and you will also learn to solve real life problems involving directional bearings.

Example 1: Solve the right triangle shown for all unknown sides and angles.

Example 2: A safety regulation states that the maximum angle of elevation for a rescue ladder is 72 °. A fire department’s longest ladder is 110 feet. What is the maximum safe rescue height?

Example 3: At a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 35 °, whereas the angle of elevation to the top is 53 °, as shown in the picture. Find the height of the smokestack alone.

Example 4: A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end. Find the depression of the bottom of the pool.

In surveying and navigation, directions can be given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fixed north-south line. For instance, the bearing S 35°E means 35 degrees east of south.

In air navigation, bearings are measured in degrees clockwise from north. Examples of air navigation bearings are shown below.

Example 5: A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 pm the ship changes course to N 54 ° W. Find the ship’s bearing and distance from the port of departure at 3 pm.

Example 6: A sailboat leaves a pier and heads due west at 8 knots. After 15 minutes the sailboat tacks, changing course to N 16°W at 10 knots. Find the sailboat’s bearing and distance from the pier after 12 minutes on this course.