Chapter 4 Trigonometry

1. Chapter 4 Trigonometry Objectives: • Convert between degree and radian measure • Pythagorean identities • Find the trigonometric values of any angle measure • Unit circle • Graph trigonometric functions • Solve for lengths and angle measures of any type of triangle. • Verify identities • Application of trigonometry

2. Uses of trigonometry: Used to describe the motion of any object that behaves in a circular, oscillating or periodic manner. Angle: Consists of two rays or half lines that originate at a common point called the vertex. These two rays have names: • Terminal side: The ray that rotates to generate the angle. • Initial side: The ray that does not move. Angles are commonly denoted using lower case Greek letters. alpha, beta, theta, gamma (respectively)

3. To better describe the formation of angles we superimpose an xy plane on the angle with the vertex at the origin. y Angles are generated by the terminal side rotating counterclockwise or clockwise. B If angles are generated by the terminal side moving counterclockwise the angle is positive. A x 0 If angles are generated by the terminal side moving clockwise the angle is negative. The direction of the arrow inside the angle will tell you if the terminal side is generating a positive or negative angle.

4. y B If angles are generated by the terminal side moving clockwise the angle is negative. A x 0

5. If we have an angle of 390°, this is one revolution of the unit circle plus 30°. So, we say that 390° is coterminal with 30° Coterminal angles differ by multiples of 360° Ex 1: Determine if the following angles are coterminal. a.) 90° and 1170 ° b.) 123 ° and 844 ° YES NO Solution: If you get a whole number they are coterminal. If the result is not a whole number then they are not coterminal.

6. To define the measure of an angle, we first add the unit circle centered at the origin to the coordinate system. y This is called a unit circle because the length of the radius is 1. 1 -1 x 1 The equation of the unit circle is: -1 The domain is: [-1, 1] To generate angles we must consider the terminal and initial sides. The initial side is aligned with the x-axis and the terminal side starts at the x-axis and rotates to generate the angle.

7. y This point is P(t). P is the function and t is the input value. x t For every P(t) on the unit circle we can define its measure by using degrees or radians. Radian Measure: For any real number t, the angle generated by rotating counterclockwise from the positive x-axis to the point P(t) on the unit circle is said to have radian measure t.

8. t 3 2 y (0,1) 1 (-1,0) (1,0) x …and so on… -1 (0,-1) -2 -3

9. We can see that an angle that measures 90° is the same measure as We can also see that 180° is We will use this fact to convert between degrees and radian measure. To convert degree measure to radian measure you multiply the degree measure by: Ex:Convert the following degree measure to radian measure. a.) 150° b.) 225° c.) -72°

10. To convert radian measure to degree measure you multiply the radian measure by: Ex 2:Convert the following radian measures to degree measure.

11. 3.2 The Sine and Cosine Functions Recall: The terminal side rotates to generate the angles. There are infinitely many points on the unit circle that the terminal side “could” generate. We will only “memorize” a few of them. We will memorize all angles on the unit circle that are in increments of 30° and 45°.

12. y Unit Circle 90° 120° 60° 45° 135° 150° 30° 180° 0° x 330° 210° 315° 225° 240° 300° 270°

13. An ordered pair has the form: (x, y) The Sine and Cosine Functions: Suppose that the coordinates of the point P(t) on the unit circle are (x(t), y(t)). Then the sine of t, written sin t, and the cosine of t, written cos t, are defined by sint t = y(t) and cos t = x(t) Our new ordered pairs are of the form: P(t) = (cos t, sin t)

14. Finding the cosine and sine values of the common angles on the unit circle. y 90° 120° 60° 45° 135° 150° 30° 180° 0° x 330° 210° 315° 225° We create right triangles by drawing lines perpendicular to the x or y axis. It does not matter which axis. 240° 300° 270°

15. To find the cosine and sine of 30° we must use the triangle we created. 1 60° 30° 1st: We know the third angle is 60° by the triangle sum theorem. 2nd: We know the length of the hypotenuse since this is a unit circle: r = 1 3rd: Using the properties of a 30-60-90 right triangle we can find the other two sides. Since the base of this triangle is on the x-axis this side would represent the cos t. The height would represent the sin t. Recall: * The side opposite the 30° angle is half of the hypotenuse. * The side opposite the 60° angle is the product of the short leg and the square root of 3.

16. If the hypotenuse has length 1, then the side opposite 30° is ½ . The side opposite the 60° angle is This is how you will solve for all sine and cosine on the unit circle dealing with 30 – 60 – 90 right triangles. So, P(t) = Finding the coordinates of a point using a 45 – 45 – 90 right triangle. We know: • The third angle is 45° 45° • The hypotenuse has length 1. 1 • In a 45 – 45 – 90 right triangle the legs are • the same length. We can call both legs • the same variable since they are equal. 45° Now solve for a.

17. We rationalized the denominator in this step because we do not leave radicals in the denominator. Since the legs have the same length, the cosine and sine values are the same.

18. Pythagorean Identity: For all real numbers t, (sin t)2 + (cos t)2 = 1 Because of the Pythagorean Identity, sine and cosine have bounds: For all real numbers t, The cosine function is even The sine function is odd For all real numbers t, cos(-t) = cos t For all real numbers t, sin(-t) = - sin t

19. A Periodic Function is a function that repeats the same thing over and over again. Trigonometric Functions are periodic because they repeat. Sine and Cosine functions have a period of From your unit circle you can see that the ordered pairs on your unit circle do not begin to repeat until after one complete revolution of the unit circle 360° Reference Number For any real number t, the reference number r associated with t is the shortest distance along the unit circle from t to the x-axis. The reference number r is always in the interval Since the answer is always in the above interval, it is Ex 2:What is the reference number of

20. Ex 3:Determine the values of t in that satisfy This is simply a quadratic function. Use what you know. You can view this quadratic as… We cannot work with it this way; we can only have one unknown. We need to turn both variables into y (sine) or x (cosine). (sin t)2 + (cos t)2 = 1 Pythagorean Identity: Manipulate this identity so that it is something that you can use. We can replace (sin t)2 with (1 – (cos t)2) Now distribute Now factor and set each factor equal to zero.

21. To make things easy, and we don’t like our leading coefficient being negative, multiply both sides by -1. Now factor It may help you to view it like this to factor. Set each factor equal to zero and solve. (2x + 1)(x – 2) = 0 2x + 1 = 0 and x – 2 = 0 Replace x with cos t since x represents cos t. Where on the unit circle is this true? cos t = 2 will never happen; it is outside the bounds of cosine.

22. Lesson 3.3: Graphs of the Sine and Cosine Functions Sine Curve We will use the values that we memorized from the unit circle to graph one period of the sine function. Sin t 1 t -1 The more points you plot the more precise your graph will be. I will plot only the 45° increments of the angles.

23. Cosine Curve cos t 1 t -1

24. Ex 1: Use the graphs of y = sin x and y = cos x to sketch the graphs of Trigonometric functions are no different than any other function when “shifts” are involved… minus to the right …plus to the left. Sin t 1 graph of y = sin x t -1 Again, we must be sure the leading coefficient is 1 before we try and see the shifts.

25. Note: Your shift will help guide you as to what you should count by on the x-axis. sin x 1 x -1 Q: What is the last value going to be that we will write on the x-axis? A: NOW YOU TRY THE SECOND ONE!!! 

26. sin x 1 x -1 You try…

27. We can see from the previous 4 graphs that when we shift one graph to the right or left that we obtain the graph of a different equation, such as: The graph of: These are all true only because we are shifting by When you shift by other increments different equalities occur.

28. When shifting by increments of pi… Ex 1: Sketch the graph of All trig functions are of the form: y = A sin(Bx + C) The name of the trig function changes only. |A| is the amplitude Don’t forget!!! The leading coefficient must be 1 before you can see the shift!!! Period is Shift is

29. Amplitude is: 2; so the height will go to +2 and -2 on y-axis. B = 1, so the period has not changed. Shift: right units First, I will graph y = 2cosx, then shift this graph to the right units. Since the period did not change and the shift is units, we will count by on the x-axis. cos x y = 2cos x Notice that the zeros of this function do not change. Amplitude is just a vertical elongation or compression of the graph. 2 x Now, we shift this graph! -2

30. cos x 2 x -2 Ex 2: Sketch the graph of B = 3, not 1 so we must find the length of the new period. Recall: period is found by: Always reduce if necessary. Instead of this graph being graphed from 0 to 2π, the entire cosine curve will be graphed between 0 and

31. Recall: To see a horizontal shift the leading coefficient must be 1 and it is not. We must factor out the 3 from the quantity. We will shift this graph to the right units. The main points of the cosine curve (y = cos x) are: cos x 2 Divide all of these orginal x-coordinates by 3 and multiply the y-coordinates by 2, these will be the locations of your new points. x -2 with shift New points:

32. Ex 3: Sketch the graph of cos x 2 x -2

33. 3.4 Other Trigonometric Functions The Tangent, Cotangent, Secant, and Cosecant Functions: The tangent, cotangent, secant and cosecant functions, written respectively as tan x, cot x, sec x, and csc x are defined by the quotients Note: Tangent and secant are only defined when cos x ≠ 0 cotangent is cosecant are only defined when sin x ≠ 0

34. + + - - + - - + + - + - + + - - + - - + + - + -

35. Ex: Determine the values of the other trigonometric functions. Solution: Since we know that csc x is the reciprocal of sin x, write the reciprocal of sin x. Next we must find cos x because the remaining trig. functions contain it. What do you know that involves both sine and cosine that will help you find cos x? Pythagorean Identity.

36. The interval for sin x is given. Quadrant II Cosine is negative in QII, so… cot x = or you could have started at the beginning to find this solution. sec x is the reciprocal of cos x. Write it and simplify: Note: You are now ready to write your answers. Make sure they have the correct sign for quadrant II.

37. Ex 2: Find the values of the other trigonometric functions. Solution: We can see that we are in QIII. The given is the cosine value for

38. The graph of the tangent function. The tangent function is zero when the sine function is zero because sine is in the numerator of the tangent function. The tangent function is undefined when the cosine function is zero because the cosine function is in the denominator of the tangent function. The tangent function will be zero at: The tangent function is undefined at: tan t 1 t y = tan x -1

39. Sine and cosine have periods of , therefore, tangent will also repeat itself on that same period. Ex 3:Sketch the following graphs: Solution: a.) B = 2; set up your inequality: The length of your new period is . Since we divided the period by 2 we will also have to divide the restrictions by 2: Original restrictions: New Restrictions:

40. a.) y = tan 2x

41. b.)

42. This graph is the graph from part b reflected over the x-axis. c.)

43. c. contd.) Now we shift it up one unit.

44. Note: The zeros from the graph are not obvious. To find them we would set the function equal to zero. Which implies… If we let x = We have… tan x = 1; where does this occur, when x = ? We will find this later this chapter. 

45. Now that we have the graphs of sine, cosine and tangent we can graph the remaining trig. Functions using the reciprocal technique. Do you remember those properties of graphing reciprocals? … as f(x) increases, its reciprocal… decreases! The Cosecant function: Graph: y = sin x. There, we have vertical asymptotes. y= csc x is undefined at: Where sin x = 0. Now, use the fact that as y = sin x increases, y = csc x decreases and vice versa.

47. Ex: Graph y = sec x

48. Ex: Graph y = cot x Be careful with y = cot x. The restrictions for y = tan x are different from y = cot x. They have different vertical asymptotes.

49. Ex: Sketch the graph of Note: There are several strategies for graphing: • You could start with the parent function and build • on it one change at a time. • You can find the most important pieces of information • of this new graph and make the changes to the pieces, • and then plot your new points and new vertical • asymptotes. This time, lets make the changes to the parent function: y = csc x

50. y = csc x has vertical asymptotes at: y = csc 3x has vertical asymptotes at: