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CHAPTER 4 Trigonometric Functions

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Page 1: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

CHAPTER 4

Trigonometric Functions

Page 2: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.1 Angles & Radian Measure• Objectives

– Recognize & use the vocabulary of angles– Use degree measure– Use radian measure– Convert between degrees & radians– Draw angles in standard position– Find coterminal angles– Find the length of a circular arc– Use linear & angular speed to describe motion on a

circular path

Page 3: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Angles

• An angle is formed when two rays have a common endpt.

• Standard position: one ray lies along the x-axis extending toward the right

• Positive angles measure counterclockwise from the x-axis

• Negative angles measure clockwise from the x-axis

Page 4: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Angle Measure

• Degrees: full circle = 360 degrees– Half-circle = 180 degrees– Right angle = 90 degrees

Radians: one radian is the measure of the central angle that intercepts an arc equal in length to the length of the radius (we can construct an angle of measure = 1 radian!)

Full circle = 2 radians

Half circle = radians

Right angle = radians

2

Page 5: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Radian Measure• The measure of the angle in radians is the ratio of the

arc length to the radius

• Recall half circle = 180 degrees= radians• This provides a conversion factor. If they are equal,

their ratio=1, so we can convert from radians to degrees (or vice versa) by multiplying by this “well-chosen one.”

• Example: convert 270 degrees to radians

r

s

2

3

180270

Page 6: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Convert 145 degrees to radians.

4

3)4

4)3

2)2

)1

Page 7: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Coterminal angles

• Angles that have rays at the same spot.

• Angle may be positive or negative (move counterclockwise or clockwise) (i.e. 70 degree angle coterminal to -290 degree angle)

• Angle may go around the circle more than once (i.e. 30 degree angle coterminal to 390 degree angle)

Page 8: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Arc length

• Since radians are defined as the central angle created when the arc length = radius length for any given circle, it makes sense to consider arc length when angle is measured in radians

• Recall theta (in radians) is the ratio of arc length to radius

• Arc length = radius x theta (in radians)

rs

Page 9: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Linear speed & Angular speed

• Speed a particle moves along an arc of the circle (v) is the linear speed (distance, s, per unit time, t)

• Speed which the angle is changing as a particle moves along an arc of the circle is the angular speed.(angle measure in radians, per unit time, t)

t

sv

t

Page 10: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Relationship between linear speed & angular speed

• Linear speed is the product of radius and angular speed.

• Example: The minute hand of a clock is 6 inches long. How fast is the tip of the hand moving?

• We know angular speed = 2 pi per 60 minutes

rv

min6.

min5min10

2

min60

26

ininininv

Page 11: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.2 Trigonometric Functions: The Unit Circle• Objectives

– Use a unit circle to define trigonometric functions of real numbers

– Recognize the domain & range of sine & cosine

– Find exact values of the trig. functions at pi/4

– Use even & odd trigonometric functions

– Recognize & use fundamental identities

– Use periodic properties

– Evaluate trig. functions with a calculator

Page 12: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

What is the unit circle?• A circle with radius = 1 unit• Why are we interested in this circle? It provides

convenient (x,y) values as we work our way around the circle.

• (1,0), theta = 0• (0,1), theta = pi/2• (-1,0), theta = pi• (0,-1), theta = 3 pi/2• ALSO, any (x,y) point on the circle would be at the

end of the hypotenuse of a right triangle that extends from the origin, such that 122 yx

Page 13: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

sin t and cos t

• For any point (x,y) found on the unit circle, x=cos t and y=sin t

• t = any real number, corresponding to the arc length of the unit circle

• Example: at the point (1,0), the cos t = 1 and sin t = 0. What is t? t is the arc length at that point AND since it’s a unit circle, we know the arc length = central angle, in radians. THUS, cos (0) = 1 and sin (0)=0

Page 14: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Relating all trigonometric functions to sin t and cos t

y

x

t

tt

xtt

ytt

x

y

t

tt

)sin(

)cos()cot(

1

)cos(

1)sec(

1

)sin(

1)csc(

)cos(

)sin()tan(

Page 15: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Pythagorean Identities

• Every point (x,y) on the unit circle corresponds to a real number, t, that represents the arc length at that point

• Since and x = cos(t) and y=sin(t), then

• If each term is divided by , the result is

• If each term is divided by , the result is

122 yx1sincos 22 tt

t2cos

tttt

t 2222

2

sectan1,cos

1

cos

sin1

t2sin

tttt

t 2222

2

csc1cot,sin

11

sin

cos

Page 16: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Given csc t = 13/12, find the values of the other 6 trig. functions of t

• sin t = 12/13 (reciprocal)

• cos t = 5/13 (Pythagorean)

• sec t = 13/5 (reciprocal)

• tan t = 12/5 (sin(t)/cos(t))

• cot t = 5/12 (reciprocal)

Page 17: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Trig. functions are periodic• sin(t) and cos(t) are the (x,y) coordinates

around the unit circle and the values repeat every time a full circle is completed

• Thus the period of both sin(t) and cos(t) = 2 pi

• sin(t)=sin(2pi + t) cos(t)=cos(2pi + t)

• Since tan(t) = sin(t)/cos(t), we find the values repeat (become periodic) after pi, thus tan(t)=tan(pi + t)

Page 18: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.3 Right Triangle Trigonometry

• Objectives

– Use right triangles to evaluate trig. Functions

– Find function values for 30 degrees, 45 degrees & 60 degrees

– Use equal cofunctions of complements

– Use right triangle trig. to solve applied problems

3,4,6

Page 19: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Within a unit circle, and right triangle can be sketched

• The point on the circle is (x,y) and the hypotenuse = 1. Therefore, the x-value is the horizontal leg and the y-value is the vertical leg of the right triangle formed.

• cos(t)=x which equals x/1, therefore the cos (t)=horizontal leg/hypotenuse = adjacent leg/hypotense

• sin(t)=y which equals y/1, therefore the sin(t) = vertical leg/hypotenuse = opposite leg/hypotenuse

Page 20: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

The relationships holds true for ALL right triangles (other 3 trig.

functions are found as reciprocals)

adjacent

opposite

hypotenuse

adjacent

hypotenuse

opposite

cos

sintan

cos

sin

Page 21: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Find the value of 6 trig. functions of the angles in a right triangle.

• Given 2 sides, the value of the 3rd side can be found, using Pythagorean theorem

• After side lengths of all 3 sides is known, find sin as opposite/hypotenuse

• cos = adjacent/hypotenuse

• tan = opposite/adjacent

• csc = 1/sin

• sec = 1/cos

• cot= 1/tan

Page 22: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Given a right triangle with hypotenuse =5 and side adjacent

angle B of length=2, find tan B

5

2)4

21

2)3

2

21)2

21)1

Page 23: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Special Triangles

• 30-60 right triangle, ratio of sides of the triangle is 1:2: , 2 (longest) is the length of the hypotenuse, the shortest side (opposite the 30 degree angle) is 1 and the remaining side (opposite the 60 degree angle) is

• 45-45 right triangle: The 2 legs are the same length since the angles opposite them are equal, thus 1:1. Using pythagorean theorem, the remaining side, the hypotenuse, is 2

3

3

Page 24: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Cofunction Identities

• Cofunctions are those that are the reciprocal functions (cofunction of tan is cot, cofunction of sin is cos, cofunction of sec is csc)

• For an acute angle, A, of a right triangle, the side opposite A would be the side adjacent to the other acute angle, B

• Therefore sin A = cos B• Since A & B are the acute angles of a right

triangle, their sum = 90 degrees, thus B=• function(A)=cofunction )90( A

A90

Page 25: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.4 Trigonometric Functions of Any Angle

• Objectives

– Use the definitions of trigonometric functions of any angle

– Use the signs of the trigonometric functions

– Find reference angles

– Use reference angles to evaluate trigonometric functions

Page 26: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Trigonometric functions of Any Angle

• Previously, we looked at the 6 trig. functions of angles in a right triangle. These angles are all acute. What about negative angles? What about obtuse angles?

• These angles exist, particularly as we consider moving around a circle

• At any point on the circle, we can drop a vertical line to the x-axis and create a triangle. Horizontal side = x, vertical side=y, hypotenuse=r.

Page 27: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Trigonometric Functions of Any Angle (continued)

• If, for example, you have an angle whose terminal side is in the 3rd quadrant, then the x & y values are both negative. The radius, r, is always a positive value.

• Given a point (-3,-4), find the 6 trig. functions associated with the angle formed by the ray containing this point.

• x=-3, y=-4, r =

• (continued next slide)

525)4()3( 22

Page 28: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Example continued

• sin A = -4/5, cos A = -3/5, tan A = 4/3

• csc A = -5/4, sec A = -5/3, cot A = ¾

• Notice that the same values of the trig. functions for angle A would be true for the angles 360+A, A-360 (negative values)

Page 29: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Examining the 4 quadrants

• Quadrant I: x & y are positive– all 6 trig. functions are positive

• Quadrant II: x negative, y positive– positive: sin, csc negative: cos, sec, tan, cot

• Quadrant III: x negative, y negative– positive: tan, cot negative: sin, csc, cos, sec

• Quadrant IV: x positive, y negative– positive: cos, sec negative: sin, csc, cot, tan

Page 30: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Reference angles

• Angles in all quadrants can be related to a “reference” angle in the 1st quadrant

• If angle A is in quadrant II, it’s related angle in quad I is 180-A. The numerical values of the 6 trig. functions will be the same, except the x (cos, sec, tan, cot) will all be negative

• If angle A is in quad III, it’s related angle in quad I is 180+A. Now x & y are both neg, so sin, csc, cos, sec are all negative.

Page 31: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Reference angles cont.

• If angle A is in quad IV, the reference angle is 360-A. The y value is negative, so the sin, csc, tan & cot are all negative.

Page 32: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Special angles

• We often work with the “special angles” of the “special triangles.” It’s good to remember them both in radians & degrees

• If you know the trig. functions of the special angles in quad I, you know them in every quadrant, by determining whether the x or y is positive or negative

290,

445,

360,

630

Page 33: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.5 Graphs of Sine & Cosine

• Objectives– Understand the graph of y = sin x– Graph variations of y = sin x– Understand the graph of y = cos x– Graph variations of y = cos x– Use vertical shifts of sin & cosine curves– Model periodic behavior

Page 34: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graphing y = sin x

• If we take all the values of sin x from the unit circle and plot them on a coordinate axis with x = angles and y = sin x, the graph is a curve

• Range: [-1,1]• Domain: (all reals)

Page 35: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graphing y = cos x

• Unwrap the unit circle, and plot all x values from the circle (the cos values) and plot on the coordinate axes, x = angle measures (in radians) and y = cos x

• Range: [-1,1]• Domain: (all reals)

Page 36: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Comparisons between y=cos x and y=sin x

• Range & Domain: SAME– range: [-1,1], domain: (all reals)

• Period: SAME (2 pi)• Intercepts: Different

– sin x : crosses through origin and intercepts the x-axis at all multiples of

– cos x: intercepts y-axis at (0,1) and intercepts x-axis at all odd multiples of

,...)3,2,,0,,2,3(....,

,...

2

3,2,2

,2

3...,

2

Page 37: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Amplitude & Period• The amplitude of sin x & cos

x is 1. The greatest distance the curves rise & fall from the axis is 1.

• The period of both functions is 2 pi. This is the distance around the unit circle.

• Can we change amplitude? Yes, if the function value (y) is multiplied by a constant, that is the NEW amplitude, example: y = 3 sin x

Page 38: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Amplitude & Period (cont)

• Can we change the period? Yes, the length of the period is a function of the x-value.

• Example: y = sin(3x)– The amplitude is still 1.

(Range: [-1,1])– Period is

3

2

Page 39: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Phase Shift

• The graph of y=sin x is “shifted” left or right of the original graph

• Change is made to the x-values, so it’s addition/subtraction to x-values.

• Example: y = sin(x- ), the graph of y=sin x is shifted right

3

3

Page 40: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Vertical Shift

• The graph y=sin x can be shifted up or down on the coordinate axis by adding to the y-value.

• Example: • y = sin x + 3 moves

the graph of sin x up 3 units.

Page 41: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph y = 2cos(x- ) - 2

• Amplitude = 2• Phase shift = right• Vertical shift = down 2

4

4

Page 42: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.6 Graphs of Other Trigonometric Functions

• Objectives– Understand the graph of y = tan x– Graph variations of y = tan x– Understand the graph of y = cot x– Graph variations of y = cot x– Understand the graphs of y = csc x and y = sec x

Page 43: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

y = tan x

• Going around the unit circle, where the y value is 0, (sin x = 0), the tangent is undefined.

• At x = the graph of y = tan x has vertical asymptotes

• x-intercepts where cos x = 0, x =

,...)2

3,2,

2,

2

3(...

,...)2,,0,,2(...

Page 44: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Characteristics of y = tan x

• Period = • Domain: (all reals except odd multiples of • Range: (all reals)• Vertical asymptotes: odd multiples of • x – intercepts: all multiples of • Odd function (symmetric through the origin, quad

I mirrors to quad III)

2

Page 45: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Transformations of y = tan x

• Shifts (vertical & phase) are done as the shifts to y = sin x

• Period change (same as to y=sin x, except the original period of tan x is pi, not 2 pi)

Page 46: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph y = -3 tan (2x) + 1

• Period is now pi/2

• Vertical shift is up 1

• -3 impacts the “amplitude”

• Since tan x has no amplitude, we consider the point ½ way between intercept & asymptote, where the y-value=1. Now the y-value at that point is -3.

• See graph next slide.

Page 47: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph y = -3 tan (2x) + 1

Page 48: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graphing y = cot x

• Vertical asymptotes are where sin x = 0, (multiples of pi)

• x-intercepts are where cos x = 0 (odd multiples of pi/2)

Page 49: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

y = csc x

• Reciprocal of y = sin x• Vertical tangents where sin x = 0 (x = integer

multiples of pi)• Range: • Domain: all reals except integer multiples of pi• Graph on next slide

),1[]1,(

Page 50: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph of y = csc x

Page 51: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

y = sec x

• Reciprocal of y = cos x• Vertical tangents where cos x = 0 (odd multiples

of pi/2)• Range: • Domain: all reals except odd multiples of pi/2• Graph next page

),1[]1,(

Page 52: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph of y = sec x

Page 53: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.7 Inverse Trigonometric Functions

• Objectives– Understand the use the inverse sine function– Understand and use the inverse cosine function– Understand and use the inverse tangent function– Use a calculator to evaluate inverse trig. functions– Find exact values of composite functions with

inverse trigonometric functions

Page 54: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

What is the inverse sin of x?

• It is the ANGLE (or real #) that has a sin value of x.• Example: the inverse sin of ½ is pi/6 (arcsin ½ = pi/6)• Why? Because the sin(pi/6)= ½• Shorthand notation for inverse sin of x is arcsin x or

• Recall that there are MANY angles that would have a sin value of ½. We want to be consistent and specific about WHICH angle we’re referring to, so we limit the range to (quad I & IV)

x1sin

2,

2

Page 55: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Find the domain of y =

• The domain of any function becomes the range of its inverse, and the range of a function becomes the domain of its inverse.

• Range of y = sin x is [-1,1], therefore the domain of the inverse sin (arcsin x) function is [-1,1]

x1sin

Page 56: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Trigonometric values for special angles

• If you know sin(pi/2) = 1, you know the inverse sin(1) = pi/2

• KNOW TRIG VALUES FOR ALL SPECIAL ANGLES (once you do, you know the inverse trigs as well!)

Page 57: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Find

4)4

4

3)3

4

7)2

4)1

2

2sin 1

Page 58: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph y = arcsin (x)

Page 59: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

The inverse cosine function

• The inverse cosine of x refers to the angle (or number) that has a cosine of x

• Inverse cosine of x is represented as arccos(x) or

• Example: arccos(1/2) = pi/3 because the cos(pi/3) = ½

• Domain: [-1,1] • Range: [0,pi] (quadrants I & II)

x1cos

Page 60: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph y = arccos (x)

Page 61: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

The inverse tangent function

• The inverse tangent of x refers to the angle (or number) that has a tangent of x

• Inverse tangent of x is represented as arctan(x) or

• Example: arctan(1) = pi/4 because the tan(pi/4)=1

• Domain: (all reals) • Range: [-pi/2,pi/2] (quadrants I & IV)

x1tan

Page 62: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Graph y = arctan(x)

Page 63: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Evaluating compositions of functions & their inverses

• Recall: The composition of a function and its inverse = x. (what the function does, its inverse undoes)

• This is true for trig. functions & their inverses, as well ( PROVIDED x is in the range of the inverse trig. function)

• Example: arcsin(sin pi/6) = pi/6, BUT arcsin(sin 5pi/6) = pi/6

• WHY? 5pi/6 is NOT in the range of arcsin x, but the angle that has the same sin in the appropriate range is pi/6

Page 64: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

4.8 Applications of Trigonometric Functions

• Objectives

– Solve a right triangle.

– Solve problems involving bearings.

– Model simple harmonic motion.

Page 65: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Solving a Right Triangle• This means find the values of all angles and all side

lengths.• Sum of angles = 180 degrees, and if one is a right

angle, the sum of the remaining angles is 90 degrees.

• All sides are related by the Pythagorean Theorem:

• Using ratio definition of trig functions (sin x = opposite/hypotenuse, tan x = opposite/adjacent, cos x = adjacent/hypotenuse), one can find remaining sides if only one side is given

222 cba

Page 66: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Example: A right triangle has an hypotenuse = 6 cm with an angle =

35 degrees. Solve the triangle.• cos(35 degrees) = .819 (using calculator)• cos(35 degrees) = adjacent/6 cm• Thus, .819 = adjacent/6 cm, adjacent = 4.9 cm• Remaining angle = 55 degrees• Remaining side:

cma

a

a

12

122436

6)9.4(2

222

Page 67: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure

Trigonometry & Bearings

• Bearings are used to describe position in navigation and surveying. Positions are described relative to a NORTH or SOUTH axis (y-axis). (Different than measuring from the standard position, the positive x-axis.)

• means the direction is 55 degrees from the north toward the east (in quadrant I)

• means the direction is 35 degrees from the south toward the west (in quadrant III)

EN 55

WS 35

Page 68: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure
Page 69: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure
Page 70: CHAPTER 4 Trigonometric Functions. 4.1 Angles & Radian Measure Objectives –Recognize & use the vocabulary of angles –Use degree measure –Use radian measure