Presentation on theme: "Trigonometry Trigonometry begins in the right triangle, but it doesn’t have to be restricted to triangles. The trigonometric functions carry the ideas."— Presentation transcript:
1TrigonometryTrigonometry begins in the right triangle, but it doesn’t have to be restricted to triangles. The trigonometric functions carry the ideas of triangle trigonometry into a broader world of real-valued functions and wave forms.
2Trigonometry Topics Radian Measure The Unit Circle Trigonometric FunctionsLarger AnglesGraphs of the Trig FunctionsTrigonometric IdentitiesSolving Trig Equations
3Radian MeasureTo talk about trigonometric functions, it is helpful to move to a different system of angle measure, called radian measure.A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle.
4Radian MeasureThere are 2 radians in a full rotation -- once around the circleThere are 360° in a full rotationTo convert from degrees to radians or radians to degrees, use the proportion
5Sample Problems Find the radian measure equivalent of 210° Find the degree measure equivalent of radians.
6The Unit CircleImagine a circle on the coordinate plane, with its center at the origin, and a radius of 1.Choose a point on the circle somewhere in quadrant I.
7The Unit CircleConnect the origin to the point, and from that point drop a perpendicular to the x-axis.This creates a right triangle with hypotenuse of 1.
8The Unit Circle is the angle of rotationThe length of its legs are the x- and y-coordinates of the chosen point.Applying the definitions of the trigonometric ratios to this triangle gives1yx
9The Unit CircleThe coordinates of the chosen point are the cosine and sine of the angle .This provides a way to define functions sin() and cos() for all real numbers .The other trigonometric functions can be defined from these.
10Trigonometric Functions is the angle of rotation1yx
11Around the CircleAs that point moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold.
12Reference AnglesThe angles whose terminal sides fall in quadrants II, III, and IV will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in quadrant I.The acute angle which produces the same values is called the reference angle.
13Reference AnglesThe reference angle is the angle between the terminal side and the nearest arm of the x-axis.The reference angle is the angle, with vertex at the origin, in the right triangle created by dropping a perpendicular from the point on the unit circle to the x-axis.
14Quadrant IIOriginal angleFor an angle, , in quadrant II, the reference angle is In quadrant II,sin() is positivecos() is negativetan() is negativeReference angle
15Quadrant III For an angle, , in quadrant III, the reference angle is Original angleFor an angle, , in quadrant III, the reference angle is - In quadrant III,sin() is negativecos() is negativetan() is positiveReference angle
16Quadrant IVFor an angle, , in quadrant IV, the reference angle is 2 In quadrant IV,sin() is negativecos() is positivetan() is negativeReference angleOriginal angle
17All Seniors Take Calculus Use the phrase “All Seniors Take Calculus” to remember the signs of the trig functions in different quadrants.AllSeniorsAll functions are positiveSine is positiveTakeCalculusTan is positiveCos is positive
22Graphs of the Trig Functions SineThe most fundamental sine wave, y = sin(x),has the graph shown.It fluctuates from 0 to a high of 1, down to –1, and back to 0, in a space of 2.
23Graphs of the Trig Functions The graph of is determined by four numbers, a, b, h, and k.The amplitude, a, tells the height of each peak and the depth of each trough.The frequency, b, tells the number of full wave patterns that are completed in a space of 2.The period of the function isThe two remaining numbers, h and k, tell the translation of the wave from the origin.
24Sample ProblemWhich of the following equations best describes the graph shown?(A) y = 3sin(2x) - 1(B) y = 2sin(4x)(C) y = 2sin(2x) - 1(D) y = 4sin(2x) - 1(E) y = 3sin(4x)-2p-1p1p2p54321-1-2-3-4-5
25Sample Problem y = 3sin(2x) - 1 Find the baseline between the high and low points.Graph is translated -1 vertically.Find height of each peak.Amplitude is 3Count number of waves in 2Frequency is 2-2p-1p1p2p54321-1-2-3-4-5y = 3sin(2x) - 1
26Graphs of the Trig Functions CosineThe graph of y = cos(x) resembles the graph of y = sin(x) but is shifted, or translated, units to the left.It fluctuates from 1to 0, down to –1,back to 0 and up to1, in a space of 2.
27Graphs of the Trig Functions The values of a, b, h, and k change the shape and location of the wave as for the sine.Amplitude a Height of each peakFrequency b Number of full wave patternsPeriod 2/b Space required to complete waveTranslation h, k Horizontal and vertical shift
28Sample ProblemWhich of the following equations best describes the graph?(A) y = 3cos(5x) + 4(B) y = 3cos(4x) + 5(C) y = 4cos(3x) + 5(D) y = 5cos(3x) + 4(E) y = 5sin(4x) + 3-2p-1p1p2p8642
29Sample Problem y = 5 cos(3x) + 4 Find the baseline Vertical translation + 4Find the height of peakAmplitude = 5Number of waves in 2Frequency =3-2p-1p1p2p8642y = 5 cos(3x) + 4
30Graphs of the Trig Functions TangentThe tangent function has a discontinuous graph, repeating in a period of .CotangentLike the tangent, cotangent is discontinuous.Discontinuities of the cotangent are units left of those for tangent.
31Graphs of the Trig Functions Secant and CosecantThe secant and cosecant functions are the reciprocals of the cosine and sine functions respectively.Imagine each graph is balancing on the peaks and troughs of its reciprocal function.
32Trigonometric Identities An identity is an equation which is true for all values of the variable.There are many trig identities that are useful in changing the appearance of an expression.The most important ones should be committed to memory.
38Solving Trig Equations Solve trigonometric equations by following these steps:If there is more than one trig function, use identities to simplifyLet a variable represent the remaining functionSolve the equation for this new variableReinsert the trig functionDetermine the argument which will produce the desired value
39Solving Trig Equations To solving trig equations:Use identities to simplifyLet variable = trig functionSolve for new variableReinsert the trig functionDetermine the argument
41Law of Sines and Cosines All these relationships are based on the assumption that the triangle is a right triangle.It is possible, however, to use trigonometry to solve for unknown sides or angles in non-right triangles.
42Law of SinesIn geometry, you learned that the largest angle of a triangle was opposite the longest side, and the smallest angle opposite the shortest side.The Law of Sines says that the ratio of a side to the sine of the opposite angle is constant throughout the triangle.
43Law of SinesIn ABC, mA = 38, mB = 42, and BC = 12 cm. Find the length of side AC.Draw a diagram to see the position of the given angles and side.BC is opposite AYou must find AC, the side opposite B.ABC
44Law of Sines .... Find the length of side AC. Use the Law of Sines with mA = 38, mB = 42, and BC = 12
45Warning The Law of Sines is useful when you know the sizes of two sides and one angle ortwo angles and one side.However, the results can be ambiguous if the given information is two sides and an angle other than the included angle (ssa).
46Law of CosinesIf you apply the Law of Cosines to a right triangle, that extra term becomes zero, leaving just the Pythagorean Theorem.The Law of Cosines is most usefulwhen you know the lengths of all three sides and need to find an angle, orwhen you two sides and the included angle.
47Law of CosinesTriangle XYZ has sides of lengths 15, 22, and 35. Find the measure of the angle C.152235C
48Law of Cosines... Find the measure of the largest angle of the triangle.221535
49Laws of Sines and Cosines Law of Sines:Law of Cosines:bACcaB