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Trigonometric Functions

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1 Trigonometric Functions
CHAPTER 4 Trigonometric Functions

2 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

3 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

4 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

5 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

6 Convert 145 degrees to radians.

7 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)

8 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)

9 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)

10 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

11 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

12 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

13 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

14 Relating all trigonometric functions to sin t and cos t

15 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

16 Given csc t = 13/12, find the values of the other 6 trig
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)

17 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)

18 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

19 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

20 The relationships holds true for ALL right triangles (other 3 trig
The relationships holds true for ALL right triangles (other 3 trig. functions are found as reciprocals)

21 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

22 Given a right triangle with hypotenuse =5 and side adjacent angle B of length=2, find tan B

23 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

24 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

25 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

26 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.

27 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)

28 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)

29 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

30 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.

31 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.

32 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

33 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

34 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)

35 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)

36 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

37 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

38 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

39 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

40 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.

41 Graph y = 2cos(x- ) - 2 Amplitude = 2 Phase shift = right
Vertical shift = down 2

42 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

43 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 =

44 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)

45 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)

46 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.

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

48 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)

49 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

50 Graph of y = csc x

51 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

52 Graph of y = sec x

53 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

54 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)

55 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]

56 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!)

57 Find

58 Graph y = arcsin (x)

59 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)

60 Graph y = arccos (x)

61 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)

62 Graph y = arctan(x)

63 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

64 4.8 Applications of Trigonometric Functions
Objectives Solve a right triangle. Solve problems involving bearings. Model simple harmonic motion.

65 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

66 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:

67 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)




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