2 4.1 Angles & Radian Measure ObjectivesRecognize & use the vocabulary of anglesUse degree measureUse radian measureConvert between degrees & radiansDraw angles in standard positionFind coterminal anglesFind the length of a circular arcUse 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 rightPositive angles measure counterclockwise from the x-axisNegative angles measure clockwise from the x-axis
4 Angle Measure Degrees: full circle = 360 degrees Half-circle = 180 degreesRight angle = 90 degreesRadians: 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 radiansHalf circle = radiansRight angle = radians
5 Radian MeasureThe measure of the angle in radians is the ratio of the arc length to the radiusRecall half circle = 180 degrees= radiansThis 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
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 lengthSince 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 radiansRecall theta (in radians) is the ratio of arc length to radiusArc 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 ObjectivesUse a unit circle to define trigonometric functions of real numbersRecognize the domain & range of sine & cosineFind exact values of the trig. functions at pi/4Use even & odd trigonometric functionsRecognize & use fundamental identitiesUse periodic propertiesEvaluate 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/2ALSO, 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 tFor any point (x,y) found on the unit circle, x=cos t and y=sin tt = any real number, corresponding to the arc length of the unit circleExample: 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 pointSince and x = cos(t) and y=sin(t), thenIf 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 tsin 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 completedThus the period of both sin(t) and cos(t) = 2 pisin(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 ObjectivesUse right triangles to evaluate trig. FunctionsFind function values for 30 degrees, 45 degrees & 60 degreesUse equal cofunctions of complementsUse 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/hypotensesin(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 theoremAfter side lengths of all 3 sides is known, find sin as opposite/hypotenusecos = adjacent/hypotenusetan = opposite/adjacentcsc = 1/sinsec = 1/coscot= 1/tan
22 Given a right triangle with hypotenuse =5 and side adjacent angle B of length=2, find tan B
23 Special Triangles30-60 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) is45-45 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, BTherefore sin A = cos BSince 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 ObjectivesUse the definitions of trigonometric functions of any angleUse the signs of the trigonometric functionsFind reference anglesUse 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 circleAt 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 = ¾Notice 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 positiveall 6 trig. functions are positiveQuadrant II: x negative, y positivepositive: sin, csc negative: cos, sec, tan, cotQuadrant III: x negative, y negativepositive: tan, cot negative: sin, csc, cos, secQuadrant IV: x positive, y negativepositive: cos, sec negative: sin, csc, cot, tan
30 Reference anglesAngles in all quadrants can be related to a “reference” angle in the 1st quadrantIf 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 negativeIf 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 anglesWe often work with the “special angles” of the “special triangles.” It’s good to remember them both in radians & degreesIf 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 xGraph variations of y = sin xUnderstand the graph of y = cos xGraph variations of y = cos xUse vertical shifts of sin & cosine curvesModel periodic behavior
34 Graphing y = sin xIf 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 curveRange: [-1,1]Domain: (all reals)
35 Graphing y = cos xUnwrap 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 xRange: [-1,1]Domain: (all reals)
36 Comparisons between y=cos x and y=sin x Range & Domain: SAMErange: [-1,1], domain: (all reals)Period: SAME (2 pi)Intercepts: Differentsin x : crosses through origin and intercepts the x-axis at all multiples ofcos x: intercepts y-axis at (0,1) and intercepts x-axis at all odd multiples of
37 Amplitude & PeriodThe 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 ShiftThe graph of y=sin x is “shifted” left or right of the original graphChange 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 ShiftThe 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 ObjectivesUnderstand the graph of y = tan xGraph variations of y = tan xUnderstand the graph of y = cot xGraph variations of y = cot xUnderstand the graphs of y = csc x and y = sec x
43 y = tan xGoing 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 asymptotesx-intercepts where cos x = 0, x =
44 Characteristics of y = tan x Period =Domain: (all reals except odd multiples ofRange: (all reals)Vertical asymptotes: odd multiples ofx – intercepts: all multiples ofOdd 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 xPeriod 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.
53 4.7 Inverse Trigonometric Functions ObjectivesUnderstand the use the inverse sine functionUnderstand and use the inverse cosine functionUnderstand and use the inverse tangent functionUse a calculator to evaluate inverse trig. functionsFind 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)= ½Shorthand notation for inverse sin of x is arcsin x orRecall 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/2KNOW TRIG VALUES FOR ALL SPECIAL ANGLES (once you do, you know the inverse trigs as well!)
59 The inverse cosine function The inverse cosine of x refers to the angle (or number) that has a cosine of xInverse cosine of x is represented as arccos(x) orExample: arccos(1/2) = pi/3 because the cos(pi/3) = ½Domain: [-1,1]Range: [0,pi] (quadrants I & II)
61 The inverse tangent function The inverse tangent of x refers to the angle (or number) that has a tangent of xInverse tangent of x is represented as arctan(x) orExample: arctan(1) = pi/4 because the tan(pi/4)=1Domain: (all reals)Range: [-pi/2,pi/2] (quadrants I & IV)
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/6WHY? 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 ObjectivesSolve 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 cmThus, .819 = adjacent/6 cm, adjacent = 4.9 cmRemaining angle = 55 degreesRemaining 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)
Your consent to our cookies if you continue to use this website.