Upload
gavin-murphy
View
238
Download
4
Tags:
Embed Size (px)
Citation preview
CHAPTER 4
Trigonometric Functions
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
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
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
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
Convert 145 degrees to radians.
4
3)4
4)3
2)2
)1
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)
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
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
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
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
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
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
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(
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
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)
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)
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
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
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
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
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
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
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
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
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.
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
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)
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
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.
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.
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
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
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)
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)
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
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
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
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
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.
Graph y = 2cos(x- ) - 2
• Amplitude = 2• Phase shift = right• Vertical shift = down 2
4
4
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
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(...
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
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)
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.
Graph y = -3 tan (2x) + 1
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)
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,(
Graph of y = csc x
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,(
Graph of y = sec x
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
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
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
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!)
Find
4)4
4
3)3
4
7)2
4)1
2
2sin 1
Graph y = arcsin (x)
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
Graph y = arccos (x)
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
Graph y = arctan(x)
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
4.8 Applications of Trigonometric Functions
• Objectives
– Solve a right triangle.
– Solve problems involving bearings.
– Model simple harmonic motion.
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
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
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