1. Preparing for the SAT II
Trigonometry

2. 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

3. 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

4. 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

5. 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

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

7. 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

8. 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

9. 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

10. 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

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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. Sample Problem y = 3sin(2x) - 15 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

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

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

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

33. 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

34. 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

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

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

37. 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