Preparing for the SAT II Trigonometry

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. ©Carolyn C. Wheater, 2000

Trigonometry Topics Radian Measure The Unit Circle Trigonometric Functions Larger Angles Graphs of the Trig Functions Trigonometric Identities Solving Trig Equations ©Carolyn C. Wheater, 2000

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 intercepted arc is equal in length to the radius of the circle. ©Carolyn C. Wheater, 2000

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 ©Carolyn C. Wheater, 2000

Sample Problems Find the degree measure equivalent of radians. Find the radian measure equivalent of 210° ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

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 ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

Trigonometric Functions  is the angle of rotation 1 y x ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

Quadrant II Original angle For an angle, , in quadrant II, the reference angle is  In quadrant II, sin() is positive cos() is negative tan() is negative Reference angle ©Carolyn C. Wheater, 2000

Quadrant III Original angle For an angle, , in quadrant III, the reference angle is - In quadrant III, sin() is negative cos() is negative tan() is positive Reference angle ©Carolyn C. Wheater, 2000

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 ©Carolyn C. Wheater, 2000 Original angle

All Star Trig Class All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants. All Star All functions are positive Sine is positive Trig Class ©Carolyn C. Wheater, 2000 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. ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

Sample Problem -2p -1p 1p 2p 5 4 3 2 1 -1 -2 -3 -4 -5 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) ©Carolyn C. Wheater, 2000

Sample Problem y = 3sin(2x) - 1 5 4 3 2 1 -1 -2 -3 -4 -5 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 ©Carolyn C. Wheater, 2000 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. ©Carolyn C. Wheater, 2000

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 Frequency b Number of full wave patterns Period 2/b Space required to complete wave Translation h, k Horizontal and vertical shift ©Carolyn C. Wheater, 2000

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 -2p -1p 1p 2p 8 6 4 2 ©Carolyn C. Wheater, 2000

Sample Problem y = 5cos(3x) + 4 Find the baseline Vertical translation + 4 Find the height of peak Amplitude = 5 Number of waves in 2 Frequency =3 -2p -1p 1p 2p 8 6 4 2 y = 5cos(3x) + 4 ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

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. ©Carolyn C. Wheater, 2000

Trigonometric Identities Reciprocal Identities Quotient Identities ©Carolyn C. Wheater, 2000

Trigonometric Identities Cofunction Identities The function of an angle = the cofunction of its complement. ©Carolyn C. Wheater, 2000

Trigonometric Identities Pythagorean Identities The fundamental Pythagorean identity Divide the first by sin2x Divide the first by cos2x ©Carolyn C. Wheater, 2000

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 ©Carolyn C. Wheater, 2000

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 ©Carolyn C. Wheater, 2000

Sample Problem Solve Use the Pythagorean identity Distribute (cos2x = 1 - sin2x) Distribute Combine like terms Order terms ©Carolyn C. Wheater, 2000

Sample Problem Solve Let t = sin x Factor and solve. ©Carolyn C. Wheater, 2000

Sample Problem Solve Replace t = sin x. t = sin(x) = ½ when So the solutions are Replace t = sin x. t = sin(x) = ½ when Solve t = sin(x) = 1 when ©Carolyn C. Wheater, 2000