1. WARM UP Find sin θ, cos θ, tan θ. Then find csc θ, sec θ and cot θ. Find b 9 41 40 θ 60° 10 b

2. MORE TRIGONOMETRIC FUNCTIONS

3. OBJECTIVES Find the quadrant in which the terminal side of an angle lies. Find the trigonometric function value of an angle, or of a rotation Find trigonometric function values for angles whose terminal sides lie on an axis. Find the reference angle of a rotation and use it to find trigonometric function values.

4. TRIGONOMETRIC FUNCTIONS Consider a rotating ray with its endpoint at the origin. The ray starts in position along the positive half of the x-axis. Counterclockwise rotations will be called negative. Note that the rotating ray and the positive half of the x-axis form an angle. Thus, we often speak of “rotations” and “angles” interchangeable. The rotating ray is often called the terminal side of the angle, and the positive half of the x-axis is called the initial side. Initial side terminal side A POSITIVE ROTATION (OR ANGLE) A NEGATIVE ROTATION (OR ANGLE)

5. MEASURES OF ROTATIONS OF ANGLES The measure of an angle, or rotation may be given in degrees. For example, a complete revolution has a measure of 360°, half a revolution has a measure of 180°, a triple revolution has a measure of 360°  3 or 1080° and so on. We also speak of angles of 90 ° or 720° or -240°.

6. TRIGONOMETRIC FUNCTIONS An angle between 0° and 90° has its terminal side in the first quadrant. An angle between 90° and 180° has its terminal angle in the second quadrant. An angle between 0° and - 90° has its terminal side in the fourth quadrant, and so on. When the measure of an angle is greater than 360° the rotating ray has gone through at least one complete revolution. For example, an angle of 375° will have the same terminal side as and angle of 15°. Thus the terminal side will be in the first quadrant. 15° 375°

7. EXAMPLES In which quadrant does the terminal side of each angle lie? 153° 2253° 3-126° 4-373° 5460° First quadrant Third quadrant Fourth quadrant Second quadrant 0° 90° 180° 270°

8. TRY THIS In which quadrant does the terminal side of each angle lie? 147° 2212° 3-43° 4-135° 5365° 6740° First quadrant Third quadrant Fourth quadrant Third quadrant First quadrant 0° 90° 180° 270° First quadrant

9. TRIGONOMETRIC FUNCTIONS OF ROTATIONS In the previous section, we worked with right triangles, so the angle θ was always less than 90°. We can use rotations to apply trigonometric functions to angles of any measure. In the previous section, we worked with right triangles, so the angle θ was always less than 90°. We can use rotations to apply trigonometric functions to angles of any measure. Consider a right triangle with one vertex on the positive x-axis. The other vertex is at R, a point of the circle whose center is at the origin and whose radius (r) is the length of the hypotenuse of the triangle. Consider a right triangle with one vertex on the positive x-axis. The other vertex is at R, a point of the circle whose center is at the origin and whose radius (r) is the length of the hypotenuse of the triangle. x R(x,y) M θ O

10. TRIGONOMETRIC FUNCTIONS OF ROTATIONS Note that three of the trigonometric functions of θ are defined as follows: Note that three of the trigonometric functions of θ are defined as follows: Sin θ = side opposite θ = y hypotenuse r hypotenuse r Cos θ = side adjacent θ = x hypotenuse r hypotenuse r Tan θ = side opposite θ = y side adjacent θ x side adjacent θ x x R(x,y) M θ O r Since x and y are coordinates of the point R, we could also define these functions as follows: Sin θ = y-coordinatecos θ = x-coordinatetan θ = y-coordinate radius radius x-coordinate radius radius x-coordinate We will use these definitions for functions of angles of any measure. Note that while x and y may be either positive, negative, or 0, r is always positive.

11. EXAMPLES Find sin θ, cos θ and tan θ for angle θ. x M θ 6 Sin θ = cos θ = tan θ = 1 Θ = 150° 2 Sin θ = cos θ = tan θ = x y y Θ = 225°

12. TRY THIS… Find sin θ, cos θ and tan θ for the angle θ shown. x θ = 330° M 2 Sin θ = cos θ = tan θ =

13. TRIGONOMETRIC FUNCTIONS IN TERMS OF x, y & r The cosecant, secant and cotangent functions can also be defined in terms of x, y and r. We find the reciprocals of the sine, cosine and tangent respectively. The cosecant, secant and cotangent functions can also be defined in terms of x, y and r. We find the reciprocals of the sine, cosine and tangent respectively. csc θ = r sec θ = r cot θ = x csc θ = r sec θ = r cot θ = x y x y y x y The values of the trigonometric functions can be positive, negative, or zero, depending on where the terminal side of the angle lies. The figure at the right shows which of the trigonometric function values are positive in each of the quadrants. The values of the trigonometric functions can be positive, negative, or zero, depending on where the terminal side of the angle lies. The figure at the right shows which of the trigonometric function values are positive in each of the quadrants. x Positive: sine, cosecant III III All positive Positive: tangent, cotangent Positive: cosine, secant IV

14. EXAMPLE Give the signs of the six trigonometric function values for a rotation of 225°. 180 < 225 < 270, so R(x, y) is in the third quadrant. The tangent and cotangent are positive, and the other four function values are negative.

15. TRY THIS… Give the signs of the six trigonometric function values for a rotation of -30°. Cosine and secant function values are positive, the other four function values are negative.

16. WARM UP In what quadrant does the terminal side of each angle lie? 1.320° 2.-175° 3.-135° 4.855° 5.230° fourth third second third

17. TERMINAL SIDE ON AN AXIS  If the terminal side of an angle falls on one of the axes, the definition of the function still apply, but in some cases functions still apply, but in some cases functions will not be defined because a denominator will be 0. notice the coordinates of the points for angles of 0°, 90°, 180°, and 720°. For example, the coordinates for an angle of 90° are x = 0 and y = r. x (0, r) (0, -r) (- r, 0) (r, 0) Example 9: Find the sine, cosine, and tangent function values for 0° and 90°. sin 0° sin 90° cos 0° cos 90° tan 0°tan 90° 90°180° 270° 380°

18. TRY THIS… Find the sine, cosine, and tangent function values for 180° and 270°. sin 180° sin 270° cos 180° cos 270° tan 180°tan 270° = 0 = -1 = 0 = -1 = 0 is undefined

19. REFERENCE ANGLES We can now determine the trigonometric values for angles in other quadrants by using the values of the functions for angles between 0° and 90°. We do so by using a reference angle. We can now determine the trigonometric values for angles in other quadrants by using the values of the functions for angles between 0° and 90°. We do so by using a reference angle. DEFINITION The reference angle for a rotation is the acute angle formed by the terminal side and the x-axis.

20. EXAMPLES Find the reference angle for θ. 10. 11. Terminal side Reference angle Θ = 115° Θ = 225° To find the measure of the acute angle formed by the terminal side and the x-axis, we subtract the measure of θ from 180°. 180 – 115 = 65 The reference angle is 65°. We are looking for the acute angle formed by the terminal side and the x-axis. We subtract 180° from 225° to get the reference angle. 225 – 180 = 45 The reference angle is 45°.

21. TRY THIS Find the reference angle for θ. a. b. Θ = 330° Θ = -150° 360 – 330 = 30 The reference angle is 30°. 180 – 150 = 30 The reference angle is 30°.

22. TERMINAL SIDE ON AN AXIS  We now use the reference angle to determine trigonometric function values. Consider, for example, an angle of 150°. The terminal side makes a 30° angle with the x-axis, since 180 – 150 = 30. As the diagram shows triangle ONR is congruent to triangle ONR’. Hence the ratios of the lengths of the sides of the two triangles are the same. x y reference angle 2 We could determine the function values directly from triangle ONR, but this is not necessary. If we remember that the sine is positive in quadrant II and that the cosine and tangent are negative, we can simply use the values for 30°, prefixing the appropriate sign. We could determine the function values directly from triangle ONR, but this is not necessary. If we remember that the sine is positive in quadrant II and that the cosine and tangent are negative, we can simply use the values for 30°, prefixing the appropriate sign. R R N 30° 2 N 180°

23. EXAMPLES 12. Find the sine, cosine, and tangent of 1320°. 1320° or 240° Θ = -150° We can subtract multiples of 360°. We do this by dividing 1320 by 360 and taking the integer part. Thus we subtract three multiples of 360. 1330 –3(360) = 240 The angle with the same terminal side is 240°. This gives us a reference angle of 60°. Since 1320° is in the third quadrant, sin 1320° = -, cos 1320° = and tan 1320° = We can add multiples of 360°. We divide 1665 by 360 and take the integer part. Thus we add four multiples of 360. -1665 – 4(360) = -225 The angle with the same terminal side is - 225°. This gives us a reference angle of 45°. Since -1665° is in the second quadrant sin -1665° =, cos -1665° =, and tan -1665° = -1 180° 60° x -1665° or -225° -180° 45° θ° x 13. Find the sine, cosine, and tangent of -1665°.

24. TRY THIS… Find the sine, cosine, and tangent of each angle. a. 2370°b. -765°c. -2340°