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Trigonometry Trigonometry 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.
Trigonometry Topics • Radian Measure • The Unit Circle • Trigonometric Functions • Larger Angles • Graphs of the Trig Functions • Trigonometric Identities • Solving Trig Equations
Radian Measure • To 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 interceptedarc is equal in length to the radius of the circle.
Radian Measure • There are 2 radians in a full rotation -- once around the circle • There are 360° in a full rotation • To convert from degrees to radians or radians to degrees, use the proportion
Sample Problems • Find the radian measure equivalent of 210° • Find the degree measure equivalent of radians.
The Unit Circle • Imagine 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.
The Unit Circle • Connect 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.
The Unit Circle is the angle of rotation • The length of its legs are the x- and y-coordinates of the chosen point. • Applying the definitions of the trigonometric ratios to this triangle gives 1 y x
The Unit Circle • The 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.
Trigonometric Functions is the angle of rotation 1 y x
Around the Circle • As that point moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold.
Reference Angles • The 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.
Reference Angles • The 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.
Quadrant II • For an angle, , in quadrant II, the reference angle is • In quadrant II, • sin() is positive • cos() is negative • tan() is negative Original angle Reference angle
Quadrant III • For an angle, , in quadrant III, the reference angle is - • In quadrant III, • sin() is negative • cos() is negative • tan() is positive Original angle Reference angle
Quadrant IV • For an angle, , in quadrant IV, the reference angle is 2 • In quadrant IV, • sin() is negative • cos() is positive • tan() is negative Reference angle Original angle
All Seniors Take Calculus • Use the phrase “All Seniors Take Calculus” to remember the signs of the trig functions in different quadrants. All Seniors All functions are positive Sine is positive Take Calculus Tan is positive Cos is positive
Graphs of the Trig Functions • Sine • The 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.
Graphs 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 is • The two remaining numbers, h and k, tell the translation of the wave from the origin.
5 4 3 2 1 -2p -1p 1p 2p -1 -2 -3 -4 -5 Sample Problem • Which 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)
5 4 3 2 1 -2p -1p 1p 2p -1 -2 -3 -4 -5 Sample Problem • Find the baseline between the high and low points. • Graph is translated -1 vertically. • Find height of each peak. • Amplitude is 3 • Count number of waves in 2 • Frequency is 2 y = 3sin(2x) - 1
Graphs of the Trig Functions • Cosine • The graph of y = cos(x) resembles the graph of y = sin(x) but is shifted, or translated, units to the left. • It fluctuates from 1 to 0, down to –1, back to 0 and up to 1, in a space of 2.
Graphs 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 peak • Frequencyb Number of full wave patterns • Period 2/bSpace required to complete wave • Translationh, kHorizontal and vertical shift
8 6 4 2 -2p -1p 1p 2p Sample Problem • Which 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
8 6 4 2 -2p -1p 1p 2p Sample Problem • Find the baseline • Vertical translation + 4 • Find the height of peak • Amplitude = 5 • Number of waves in 2 • Frequency =3 y = 5 cos(3x) + 4
Graphs of the Trig Functions • Tangent • The tangent function has a discontinuous graph, repeating in a period of . • Cotangent • Like the tangent, cotangent is discontinuous. • Discontinuities of the cotangent are units left of those for tangent.
Graphs of the Trig Functions • Secant and Cosecant • The 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.
Trigonometric 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.
Trigonometric Identities • Quotient Identities • Reciprocal Identities
Trigonometric Identities • Cofunction Identities • The functionof an angle = the cofunction of its complement.
Trigonometric Identities • Pythagorean Identities • The fundamental Pythagorean identity • Divide the first by sin2x • Divide the first by cos2x
Solving Trig Equations • Solve trigonometric equations by following these steps: • If there is more than one trig function, use identities to simplify • Let a variable represent the remaining function • Solve the equation for this new variable • Reinsert the trig function • Determine the argument which will produce the desired value
Solving Trig Equations • To solving trig equations: • Use identities to simplify • Let variable = trig function • Solve for new variable • Reinsert the trig function • Determine the argument
Sample Problem • Solve
Law 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.
Law of Sines • In 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.
C B A Law of Sines • In 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 A • You must find AC, the side opposite B.
Law of Sines • .... Find the length of side AC. • Use the Law of Sines with mA = 38, mB = 42, and BC = 12
Warning • The Law of Sines is useful when you know • the sizes of two sides and one angle or • two 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).
Law of Cosines • If 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 useful • when you know the lengths of all three sides and need to find an angle, or • when you two sides and the included angle.
22 C 15 35 Law of Cosines • Triangle XYZ has sides of lengths 15, 22, and 35. Find the measure of the angle C.
Law of Cosines • ... Find the measure of the largest angle of the triangle. 22 15 35
Laws of Sines and Cosines • Law of Sines: • Law of Cosines: b A C c a B