1. The World’s Largest Ferris Wheel In London, British Airways built a ferris wheel which measures 450 feet in diameter. It turns continuously at a constant rate, completing a single rotation every 30 minutes. The values of f(t), your height above the ground t minutes after boarding the wheel, are shown in the table below: The graph of f(t) is shown on the next slide. t, min 0 7.5 15 22.5 30 37.5 45 52.5 60 f(t), feet 0 225 450 225 0 225 450 225 0

2. A function f is periodic if f(t+c) = f(t) for all t in the domain of f. The smallest positive constant c for which this relationship holds is called the period of f. A part of the graph of f as t varies over an interval of one period length is called a cycle. The midline of a periodic function is the horizontal line midway between the function’s maximum and minimum values. The amplitude is the vertical distance between the function’s maximum (or minimum) and the midline. Period = 30 Midline: y = 225 Amplitude = 225

3. The Unit Circle The unit circle is the circle of radius one that is centered at the origin. The equation of the unit circle is x 2 + y 2 = 1. Angles (denoted by in the figure below) can be used to locate points on the unit circle. These angles are measured counterclockwise from the positive x-axis. (1,0) (-1,0) (0,-1) (0,1) P = (x,y) x y

4. Definition of Sine and Cosine Suppose P = (x,y) is the point on the unit circle specified by the angle. We define the functions, cosine of, or, and the sine of, or, by the formulas See figure below.

5. Values of the Sine and Cosine Functions Find the values of: In each case, find the point on the unit circle determined by the given angle. The coordinates of this point are the values we seek. We have that Similarly, However, in order to evaluate we may use a calculator. We find that

6. Conventions for Working with Angles We measure angles with respect to the horizontal, not the vertical, so that 0º describes the 3 o’clock position. Positive angles are measured in the counterclockwise direction, negative angles in the clockwise direction. Large angles (greater than 360º or less than –360º) wrap around a circle more than once.

7. Definition of Sine and Cosine in Right Triangles Suppose P = (x,y) is the point on the unit circle specified by the angle. We define the functions, cosine of, or, and the sine of, or, by either pair of formulas: (-1,0) Hypotenuse Side Opposite Side Adjacent

8. Example: sine and cosine of 45º Consider an isosceles right triangle with sides equal 1. From elementary geometry, we know that the acute angles of this triangle both equal 45º, and by Pythagoras’ theorem the length of the hypotenuse is Therefore, using the right triangle definition, we have 1 1

9. Values of Sine and Cosine The values for sine and cosine in the following table of “special” angles (see pg 314 of text) should be memorized: You should be able to use reference angles (see next slide) to find the values of sine and cosine for angles which are not in the first quadrant. For example,

10. Reference Angles For an angle corresponding to the point P on the unit circle, the reference angle of is the angle between the line joining P to the origin and the nearest part of the x-axis. See Page 314 of the text. A reference angle is always between Example. Find the reference angle of x y

11. The Unit Circle x y For any angle  : (x,y) = (cos , sin  )

12. Graph of sine as a function of angle t Domain of sine is all real numbers Range of sine is Sine is an odd function—it has symmetry about the origin The period of sine is 360º.

13. Graph of cosine as a function of angle t Domain of cosine is all real numbers Range of cosine is Cosine is an even function—it has symmetry about the y-axis The period of cosine is 360º.

14. Coordinates of a point on a circle of radius r. The coordinates (x,y) of the point Q in the figure below are given by:

15. Height on the ferris wheel as a function of angle The ferris wheel described earlier has a radius of 225 feet. Find your height above the ground as a function of the angle t measured from the 3 o’clock position. P = (x, y) y x

16. Graph of ferris wheel function The graph of the function giving your height, y = f(t), in feet, as a function of the angle t, measured in degrees from the 3 o’clock position, is shown next. Period = 360 º Midline y = 225 ft. f(t)=225+225sin t

17. Definition of the tangent function Suppose P = (x, y) is the point on the unit circle specified by the angle. We define the tangent of, or tan, by tan = y/x, or by: Side Opposite Side Adjacent

18. Values and graph of the tangent function The values of the tangent function for “special” angles are: The function tan t is periodic with period 180º, and its graph is next. It has a vertical asymptote when t is an odd multiple of 90º.

19. Solving Right Triangles One of the angles of a right triangle is 90°. Thus, one part is already known and the only additional information necessary is either two sides or an acute angle and a side. With this additional information, it is possible to solve for the remaining parts. Making a carefully labeled sketch of the triangle is important in solving these problems. Problem 25, page 296. The top of a 200-foot tower is to be anchored by cables that make an angle of 30° with the ground. How long must the cables be? How far from the base of the tower should the anchors be placed? sin 30° = 200/h => h = 200/sin 30° = 400 ft tan 30° = 200/x => x = 200/tan 30° = 346.4 ft 200 ft h x 30°

20. Right Triangles: Inverse Trigonometric Functions If we know the sides of a right triangle, what are the angles? Example. Consider the 3-4-5 right triangle. Value of Ө? 3 4 5 Ө y = sin Ө Ө y Ө=36.87º

21. Definitions of Inverse Trigonometric Functions arcsin x = sin –1 x = the angle in a right triangle whose sine is x arccos x = cos –1 x = the angle in a right triangle whose cosine is x arctan x = tan -1 x = the angle in a right triangle whose tangent is x This means that for an acute angle Ө in a right triangle, sin Ө = x means Ө = sin –1 x cos Ө = x means Ө = cos –1 x tan Ө = x means Ө = tan –1 x

22. Example problem for inverse trig functions The grade of a road is 5.8%. What angle does the road make with the horizontal? We have tan Ө = 5.8/100 = 0.058. Using a calculator in degree mode, we have Ө = tan –1 (0.058) = 3.319º. 5.8 ft 100 ft Ө

23. Solving Non-right Triangles Suppose the triangle to be solved does not contain a right angle. If we are given three parts of the triangle, not all angles, then we can solve for the remaining parts. The solution exists and is unique except in one case--discussed in a later slide. Usually we label the angles of the triangle as A, B, C and the sides opposite these angles as a, b, and c, respectively. Law of Cosines: Law of Sines: ba c A C B

24. An example using the Law of Cosines A person leaves her home and walks 5 miles due east and then 3 miles northeast. How far away from home (as the crow flies) is she? As seen from her home, what is the angle  between the easterly direction and the direction to her destination? By applying the Law of Cosines, we can solve for x: A second application of the Law of Cosines gives: Home Destination x 5 3 45°135° N

25. An example using the Law of Sines Problem 35, page 306. Two fire stations are located 56.7 miles apart, at points A and B. There is a forest fire at point C. If  CAB = 54° and  CBA = 58°, which fire station is closer? How much closer? The fire station at point B is closer to the fire by 51.861 – 49.474 = 2.387 miles. 54° 68° 58° C AB b c = 56.7 miles a

26. The ambiguous case for solving a triangle Suppose we are given two sides of a triangle and the angle opposite one of them, and we are asked to find the remaining parts of the triangle. Depending on the data given, there may be: two possible triangles, one possible triangle, or no possible triangle. Example.  CAB = 45°, b = 1, and various values of a: No triangle exists. b=1 45° a=  2/2 b=1 a=3/10 One triangle exists. A b=1 45° a=8/10 A Two triangles exist. b=1 45° A a=11/10 45° A C CC B B C

27. Summary for Trigonometry in Circles and Triangles Periodic function, period, midline, and amplitude were defined. Sine, cosine, and tangent were defined in terms of the unit circle and in terms of right triangles (SOHCAHTOA). Do not use SOHCAHTOA to solve non-right triangles! If (x,y) is a point on a circle of radius r specified by angle, then Values of sine, cosine, and tangent for “special” angles were given. Reference angles can be used to find values of sine, cosine, and tangent for angles which are not in the first quadrant. The sine and cosine functions are periodic with period 360º and their domains, ranges, and graphical symmetries were given. The function tan t is periodic with period 180º and it has vertical asymptotes when t is an odd multiple of 90º.

28. Summary for Trigonometry in Circles and Triangles, cont’d. The definitions of the trig functions and the inverses of the trig functions were used in solving right triangles. The law of cosines was given, and it is a generalization of Pythagoras’ theorem. Also, the law of sines was given. The use of the law of cosines and the law of sines in solving non-right triangles was discussed.