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

Basic Terminology Quadrantal angle: Angle in standard position that doesn’t lie in any quadrant Lies in quadrant II Quadrantal angle Lies in quadrant IV

Measuring Angles Two usual ways of measuring Degrees Radians 360± in one rotation Radians 2¼ radians in one rotation

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±3502700 Answer: Example. Convert to the D±M0S00 form Problem: 73.582±

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

Radians IMPORTANT! Radians are dimensionless 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 1 revolution = 2¼ radians = 360± 1± = radians

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

Exact Values for Quadrantal Angles Quadrantal angles correspond to integer multiples of 90± or of radians

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = 0 = 0± Answer:

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = = 90± Answer:

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = ¼ = 180± Answer:

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = = 270± Answer:

Exact Values for Quadrantal Angles

Exact Values for Quadrantal Angles 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

Circles of Radius r Theorem. 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

Circles of Radius r Example. 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