# Trigonometric Functions

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Trigonometric Functions
Chapter 5 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

Angles and Their Measure
Section 5.1

Basic Terminology Ray: A half-line starting at a vertex V
Angle: Two rays with a common vertex

Basic Terminology Initial side and terminal side: The rays in an angle
Angle shows direction and amount of rotation Lower-case Greek letters denote angles

Basic Terminology Positive angle: Counterclockwise rotation
Negative angle: Clockwise rotation Coterminal angles: Share initial and terminal sides Positive angle Negative angle Positive angle

Basic Terminology Standard position: Vertex at origin
Initial side is positive x-axis

Measuring Angles Two usual ways of measuring Degrees Radians

Measuring Angles Right angle: A quarter revolution
A right angle contains 90± radians

Measuring Angles Straight angle: A half revolution.
A straight angle contains: 180± ¼ radians

Measuring Angles Negative angles have negative measure
Multiple revolutions are allowed

Degrees, Minutes and Seconds
One complete revolution = 360± One minute: One-sixtieth of a degree One minute is denoted 10 1± = 600 One second: One-sixtieth of a minute One second is denoted 100 10 = 6000

Degrees, Minutes and Seconds
Example. Convert to a decimal in degrees Problem: 64± Answer: Example. Convert to the D±M0S00 form Problem: ±

Radians Central angle: An angle whose vertex is at the center of a circle Central angles subtend an arc on the circle

Radians One radian is the measure of an angle which subtends an arc with length equal to the radius of the circle

If an angle appears with no units, it must be assumed to be in radians

Arc Length Theorem. [Arc Length] WARNING!
For a circle of radius r, a central angle of µ radians subtends an arc whose length s is s = rµ WARNING! The angle must be given in radians

Arc Length Example. Problem: Find the length of the arc of a circle of radius 5 centimeters subtended by a central angle of 1.4 radians Answer:

Radians vs. Degrees Example. Convert each angle in degrees to radians and each angle in radians to degrees (a) Problem: 45± Answer: (b) Problem: {270± (c) Problem: 2 radians

Radians vs. Degrees Measurements of common angles

Area of a Sector of a Circle
Theorem. [Area of a Sector] The area A of the sector of a circle of radius r formed by a central angle of µ radians is

Area of a Sector of a Circle
Example. Problem: Find the area of the sector of a circle of radius 3 meters formed by an angle of 45±. Round your answer to two decimal places. Answer: WARNING! The angle again must be given in radians

Linear and Angular Speed
Object moving around a circle or radius r at a constant speed Linear speed: Distance traveled divided by elapsed time t = time µ = central angle swept out in time t s = rµ = arc length = distance traveled

Linear and Angular Speed
Object moving around a circle or radius r at a constant speed Angular speed: Angle swept out divided by elapsed time Linear and angular speeds are related v = r!

Linear and Angular Speed
Example. A neighborhood carnival has a Ferris wheel whose radius is 50 feet. You measure the time it takes for one revolution to be 90 seconds. (a) Problem: What is the linear speed (in feet per second) of this Ferris wheel? Answer: (b) Problem: What is the angular speed (in radians per second)?

Key Points Basic Terminology Measuring Angles
Degrees, Minutes and Seconds Radians Arc Length Radians vs. Degrees Area of a Sector of a Circle Linear and Angular Speed

Trigonometric Functions: Unit Circle Approach
Section 5.2

Unit Circle Unit circle: Circle with radius 1 centered at the origin
Equation: x2 + y2 = 1 Circumference: 2¼

Unit Circle Travel t units around circle, starting from the point (1,0), ending at the point P = (x, y) The point P = (x, y) is used to define the trigonometric functions of t

Trigonometric Functions
Let t be a real number and P = (x, y) the point on the unit circle corresponding to t: Sine function: y-coordinate of P sin t = y Cosine function: x-coordinate of P cos t = x Tangent function: if x  0

Trigonometric Functions
Let t be a real number and P = (x, y) the point on the unit circle corresponding to t: Cosecant function: if y  0 Secant function: if x  0 Cotangent function: if y  0

Exact Values Using Points on the Circle
A point on the unit circle will satisfy the equation x2 + y2 = 1 Use this information together with the definitions of the trigonometric functions.

Exact Values Using Points on the Circle
Example. Let t be a real number and P = the point on the unit circle that corresponds to t. Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t Answer:

Trigonometric Functions of Angles
Convert between arc length and angles on unit circle Use angle µ to define trigonometric functions of the angle µ

Example. Find the values of the trigonometric functions of µ Problem: µ = 0 = 0± Answer:

Example. Find the values of the trigonometric functions of µ Problem: µ = = 90± Answer:

Example. Find the values of the trigonometric functions of µ Problem: µ = ¼ = 180± Answer:

Example. Find the values of the trigonometric functions of µ Problem: µ = = 270± Answer:

Example. Find the exact values of (a) Problem: sin({90±) Answer: (b) Problem: cos(5¼)

Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µ Problem: µ = = 45± Answer:

Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µ Problem: µ = = 60± Answer:

Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µ Problem: µ = = 30± Answer:

Exact Values for Standard Angles

Exact Values for Standard Angles
Example. Find the values of the following expressions (a) Problem: sin(315±) Answer: (b) Problem: cos({120±) (c) Problem:

Approximating Values Using a Calculator
IMPORTANT! Be sure that your calculator is in the correct mode. Use the basic trigonometric facts:

Approximating Values Using a Calculator
Example. Use a calculator to find the approximate values of the following. Express your answers rounded to two decimal places. (a) Problem: sin 57± Answer: (b) Problem: cot {153± (c) Problem: sec 2

For an angle µ in standard position, let P = (x, y) be the point on the terminal side of µ that is also on the circle x2 + y2 = r2. Then

Problem: Find the exact values of each of the trigonometric functions of an angle µ if ({12, {5) is a point on its terminal side. Answer:

Key Points Unit Circle Trigonometric Functions
Exact Values Using Points on the Circle Trigonometric Functions of Angles Exact Values for Quadrantal Angles Exact Values for Standard Angles Approximating Values Using a Calculator

Key Points (cont.) Circles of Radius r

Properties of the Trigonometric Functions
Section 5.3

Domains of Trigonometric Functions
Domain of sine and cosine functions is the set of all real numbers Domain of tangent and secant functions is the set of all real numbers, except odd integer multiples of = 90± Domain of cotangent and cosecant functions is the set of all real numbers, except integer multiples of ¼ = 180±

Ranges of Trigonometric Functions
Sine and cosine have range [{1, 1] {1 · sin µ · 1; jsin µj · 1 {1 · cos µ · 1; jcos µj · 1 Range of cosecant and secant is ({1, {1] [ [1, 1) jcsc µj ¸ 1 jsec µj ¸ 1 Range of tangent and cotangent functions is the set of all real numbers

Periods of Trigonometric Functions
Periodic function: A function f with a positive number p such that whenever µ is in the domain of f, so is µ + p, and f(µ + p) = f(µ) (Fundamental) period of f: smallest such number p, if it exists

Periods of Trigonometric Functions
Periodic Properties: sin(µ + 2¼) = sin µ cos(µ + 2¼) = cos µ tan(µ + ¼) = tan µ csc(µ + 2¼) = csc µ sec(µ + 2¼) = sec µ cot(µ + ¼) = cot µ Sine, cosine, cosecant and secant have period 2¼ Tangent and cotangent have period ¼

Periods of Trigonometric Functions
Example. Find the exact values of (a) Problem: sin(7¼) Answer: (b) Problem: (c) Problem:

Signs of the Trigonometric Functions
P = (x, y) corresponding to angle µ Definitions of functions, where defined Find the signs of the functions Quadrant I: x > 0, y > 0 Quadrant II: x < 0, y > 0 Quadrant III: x < 0, y < 0 Quadrant IV: x > 0, y < 0

Signs of the Trigonometric Functions

Signs of the Trigonometric Functions
Example: Problem: If sin µ < 0 and cos µ > 0, name the quadrant in which the angle µ lies Answer:

Quotient Identities P = (x, y) corresponding to angle µ:
Get quotient identities:

Quotient Identities Example.
Problem: Given and , find the exact values of the four remaining trigonometric functions of µ using identities. Answer:

Pythagorean Identities
Unit circle: x2 + y2 = 1 (sin µ)2 + (cos µ)2 = 1 sin2 µ + cos2 µ = 1 tan2 µ + 1 = sec2 µ 1 + cot2 µ = csc2 µ

Pythagorean Identities
Example. Find the exact values of each expression. Do not use a calculator (a) Problem: cos 20± sec 20± Answer: (b) Problem: tan2 25± { sec2 25±

Pythagorean Identities
Example. Problem: Given that and that µ is in Quadrant II, find cos µ. Answer:

Even-Odd Properties A function f is even if f({µ) = f(µ) for all µ in the domain of f A function f is odd if f({µ) = {f(µ) for all µ in the domain of f

Even-Odd Properties Theorem. [Even-Odd Properties]
sin({µ) = {sin(µ) cos({µ) = cos(µ) tan({µ) = {tan(µ) csc({µ) = {csc(µ) sec({µ) = sec(µ) cot({µ) = {cot(µ) Cosine and secant are even functions The other functions are odd functions

Even-Odd Properties Example. Find the exact values of
(a) Problem: sin({30±) Answer: (b) Problem: (c) Problem:

Fundamental Trigonometric Identities
Quotient Identities Reciprocal Identities Pythagorean Identities Even-Odd Identities

Key Points Domains of Trigonometric Functions
Ranges of Trigonometric Functions Periods of Trigonometric Functions Signs of the Trigonometric Functions Quotient Identities Pythagorean Identities Even-Odd Properties Fundamental Trigonometric Identities

Graphs of the Sine and Cosine Functions
Section 5.4

Graphing Trigonometric Functions
Graph in xy-plane Write functions as y = f(x) = sin x y = f(x) = cos x y = f(x) = tan x y = f(x) = csc x y = f(x) = sec x y = f(x) = cot x Variable x is an angle, measured in radians Can be any real number

Graphing the Sine Function
Periodicity: Only need to graph on interval [0, 2¼] (One cycle) Plot points and graph

Properties of the Sine Function
Domain: All real numbers Range: [{1, 1] Odd function Periodic, period 2¼ x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, … y-intercept: 0 Maximum value: y = 1, occurring at Minimum value: y = {1, occurring at

Transformations of the Graph of the Sine Functions
Example. Problem: Use the graph of y = sin x to graph Answer:

Graphing the Cosine Function
Periodicity: Again, only need to graph on interval [0, 2¼] (One cycle) Plot points and graph

Properties of the Cosine Function
Domain: All real numbers Range: [{1, 1] Even function Periodic, period 2¼ x-intercepts: y-intercept: 1 Maximum value: y = 1, occurring at x = …, {2¼, 0, 2¼, 4¼, 6¼, … Minimum value: y = {1, occurring at x = …, {¼, ¼, 3¼, 5¼, …

Transformations of the Graph of the Cosine Functions
Example. Problem: Use the graph of y = cos x to graph Answer:

Sinusoidal Graphs Graphs of sine and cosine functions appear to be translations of each other Graphs are called sinusoidal Conjecture.

Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj Number jAj is the amplitude

Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj Number jAj is the amplitude

Amplitude and Period of Sinusoidal Functions
Period of y = sin(!x) and y = cos(!x) is

Amplitude and Period of Sinusoidal Functions
Cycle: One period of y = sin(!x) or y = cos(!x)

Amplitude and Period of Sinusoidal Functions
Cycle: One period of y = sin(!x) or y = cos(!x)

Amplitude and Period of Sinusoidal Functions
Theorem. If ! > 0, the amplitude and period of y = Asin(!x) and y = Acos(! x) are given by Amplitude = j Aj Period =

Amplitude and Period of Sinusoidal Functions
Example. Problem: Determine the amplitude and period of y = {2cos(¼x) Answer:

Graphing Sinusoidal Functions
One cycle contains four important subintervals For y = sin x and y = cos x these are Gives five key points on graph

Graphing Sinusoidal Functions
Example. Problem: Graph y = {3cos(2x) Answer:

Finding Equations for Sinusoidal Graphs
Example. Problem: Find an equation for the graph. Answer:

Key Points Graphing Trigonometric Functions Graphing the Sine Function
Properties of the Sine Function Transformations of the Graph of the Sine Functions Graphing the Cosine Function Properties of the Cosine Function Transformations of the Graph of the Cosine Functions

Key Points (cont.) Sinusoidal Graphs
Amplitude and Period of Sinusoidal Functions Graphing Sinusoidal Functions Finding Equations for Sinusoidal Graphs

Graphs of the Tangent, Cotangent, Cosecant and Secant Functions
Section 5.5

Graphing the Tangent Function
Periodicity: Only need to graph on interval [0, ¼] Plot points and graph

Properties of the Tangent Function
Domain: All real numbers, except odd multiples of Range: All real numbers Odd function Periodic, period ¼ x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, … y-intercept: 0 Asymptotes occur at

Transformations of the Graph of the Tangent Functions
Example. Problem: Use the graph of y = tan x to graph Answer:

Graphing the Cotangent Function
Periodicity: Only need to graph on interval [0, ¼]

Graphing the Cosecant and Secant Functions
Use reciprocal identities Graph of y = csc x

Graphing the Cosecant and Secant Functions
Use reciprocal identities Graph of y = sec x

Key Points Graphing the Tangent Function
Properties of the Tangent Function Transformations of the Graph of the Tangent Functions Graphing the Cotangent Function Graphing the Cosecant and Secant Functions

Phase Shifts; Sinusoidal Curve Fitting
Section 5.6

Graphing Sinusoidal Functions
y = A sin(!x), ! > 0 Amplitude jAj Period y = A sin(!x { Á) Phase shift Phase shift indicates amount of shift To right if Á > 0 To left if Á < 0

Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á): Determine amplitude jAj Determine period Determine starting point of one cycle: Determine ending point of one cycle:

Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á): Divide interval into four subintervals, each with length Use endpoints of subintervals to find the five key points Fill in one cycle

Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á): Extend the graph in each direction to make it complete

Graphing Sinusoidal Functions
Example. For the equation (a) Problem: Find the amplitude Answer: (b) Problem: Find the period (c) Problem: Find the phase shift

Finding a Sinusoidal Function from Data
Example. An experiment in a wind tunnel generates cyclic waves. The following data is collected for 52 seconds. Let v represent the wind speed in feet per second and let x represent the time in seconds. Time (in seconds), x Wind speed (in feet per second), v 21 12 42 26 67 41 40 52 20

Finding a Sinusoidal Function from Data
Example. (cont.) Problem: Write a sine equation that represents the data Answer:

Key Points Graphing Sinusoidal Functions
Finding a Sinusoidal Function from Data