By Y. Ath

1. By Y. Ath

2. Section 1Angles

3. Basic Terminology Line AB. Line segment AB Ray AB

4. Math symbols (Greek alphabets)

5. An angle’s measure is generated by a rotation about the vertex. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle. Basic Terminology

6. Positive angle: The rotation of the terminal side of an angle is counterclockwise. Negative angle: The rotation of the terminal side is clockwise. Basic Terminology

7. Measuring Angle • Two units for measuring angle: • degrees, minutes, seconds (DMS) or decimal degrees • radians

8. The most common unit for measuring angles is the degree. of complete rotation gives an angle whose measure is 1°. Degree Measure A complete rotation of a ray gives an angle whose measure is 360°.

9. Example Convert to degrees, minutes, seconds Convert to radians

10. Example Convert to decimal degrees

11. Angles are classified by their measures. Degree Measure

12. Complementary, Supplementary Angles

13. FINDING THE COMPLEMENT AND THE SUPPLEMENT OF AN ANGLE For an angle measuring 40°, find the measure of (a) itscomplement and (b) itssupplement. Example 1 (a) To find the measure of its complement, subtract the measure of the angle from 90°. Complement of 40° (b) To find the measure of its supplement, subtract the measure of the angle from 180°. Supplement of 40°

14. FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES Find the measure of each marked angle. Example 2 Since the two angles form a right angle, they are complementary. Combine like terms. Divide by 9. Determine the measure of each angle by substituting 10 for x:

15. FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES (continued) Find the measure of each marked angle. Example 2 The angle measures are and . Since the two angles form a straight angle, they are supplementary.

16. Quadrantal Angles

17. Coterminal Angles Angles are coterminal if their initial and terminal sides are the same. Example

18. To find an expression that will generate all angles coterminal with a given angle, add integer multiples of 360° to the given angle. For example, the expression for all angles coterminal with 60° is Coterminal Angles

19. Coterminal Angles

20. ANALYZING THE REVOLUTIONS OF A CD PLAYER Example Each revolution is 360°, so a point on the edge of the CD will revolve in 2 sec. CD players always spin at the same speed. Suppose a player makes 480 revolutions per min. Through how many degrees will a point on the edge of a CD move in 2 sec? The player revolves 480 times in 1 min or times = 8 times per sec. In 2 sec, the player will revolve times.

21. Section 1.2 Angles Relationships and Similar Angles

22. Vertical Angles Alternate interior Angles Alternate exterior Angles Corresponding Angles Parallel Lines

23. Find the measures of angles 1, 2, 3, and 4, given that lines m and n are parallel. Example FINDING ANGLE MEASURES Angles 1 and 4 are alternate exterior angles, so they are equal. Subtract 3x. Add 40. Divide by 2. Angle 1 has measure Substitute 21 for x.

24. Example Angle 2 is the supplement of a 65° angle, so it has measure . FINDING ANGLE MEASURES (continued) Angle 4 has measure Substitute 21 for x. Angle 3 is a vertical angle to angle 1, so its measure is 65°.

25. Angle Sum of a Triangle The sum of the measures of the angles of any triangle is 180°.

26. Example APPLYING THE ANGLE SUM OF A TRIANGLE PROPERTY The measures of two of the angles of a triangle are 48 and 61. Find the measure of the third angle, x. The sum of the angles is 180°. Add. Subtract 109°. The third angle of the triangle measures 71°.

27. Types of Triangles Acute triangle Right triangle Obtuse triangle Equilateral triangle Isosceles triangle Scalene triangle Three equal angles Three equal sides No equal sides No equal angles Two equal sides Two equal angles

28. Proportion, Similar Triangles Proportion: One ratio or one fraction equals another Similar Triangles: Two triangles are similar if and only if corresponding sides are in proportion and the corresponding angles are congruent.

29. Example FINDING ANGLE MEASURES IN SIMILAR TRIANGLES In the figure, triangles ABC and NMP are similar.Find the measures of angles B and C. Since the triangles are similar, corresponding angles have the same measure. C corresponds to P, so angle C measures 104°. B corresponds to M, so angle B measures 31°.

30. Example DF corresponds to AB, and DE corresponds to AC, so FINDING SIDE LENGTHS IN SIMILAR TRIANGLES Given that triangle ABC and triangle DFE are similar, find the lengths of the unknown sides of triangle DFE. Similar triangles have corresponding sides in proportion.

31. Example EF corresponds to CB, so FINDING SIDE LENGTHS IN SIMILAR TRIANGLES (continued) Side DF has length 12. Side EF has length 16.

32. Sections 1.3 & 1.4 Trigonometric Functions

33. Six Trigonometric Functions Given a Point Try to remember SOH-CAH-TOA

34. The terminal side of angle  in standard position passes through the point (8, 15). Find the values of the six trigonometric functions of angle . Example FINDING FUNCTION VALUES OF AN ANGLE

35. Example We can now find the values of the six trigonometric functions of angle . FINDING FUNCTION VALUES OF AN ANGLE (continued)

36. Example FINDING FUNCTION VALUES OF AN ANGLE (continued)

37. Function values of quadrantal angles can be found with a calculator that has trigonometric function keys. Make sure the calculator is set in degree mode. Using a Calculator

38. Caution One of the most common errors involving calculators in trigonometry occurs when the calculator is set for radian measure, rather than degree measure. Be sure you know how to set your calculator in degree mode.

39. Signs of Function Values

40. Pythagorean Theorem Pythagorean Identities

41. Quotient Identities

42. FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Suppose that angle  is in quadrant II and Find the values of the other five trigonometric functions. Example Choose any point on the terminal side of angle . Let r = 3. Then y = 2. Since  is in quadrant II,

43. FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) Example Remember to rationalize the denominator.

44. FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) Example

45. USING IDENTITIES TO FIND FUNCTION VALUES Example Choose the positive square root since sin θ >0.

46. USING IDENTITIES TO FIND FUNCTION VALUES (continued) Example To find tan θ, use the quotient identity

47. USING IDENTITIES TO FIND FUNCTION VALUES Example Find sin θ and cos θ, given that and θ is in quadrant III. Since θ is in quadrant III, sin θ and cos θ will both be negative. It is tempting to say that since and then sin θ = –4 and cos θ = –3. This is incorrect, however, since both sin θ and cos θ must be in the interval [–1,1].

48. USING IDENTITIES TO FIND FUNCTION VALUES (continued) Example Use the identity to find sec θ. Then use the reciprocal identity to find cos θ. Choose the negative square root since sec θ <0 when θ is in quadrant III. Secant and cosine are reciprocals.

49. USING IDENTITIES TO FIND FUNCTION VALUES (continued) Example Choose the negative square root since sin θ <0 for θ in quadrant III.

50. USING IDENTITIES TO FIND FUNCTION VALUES (continued) Example This example can also be worked by sketching θ in standard position in quadrant III, finding r to be 5, and then using the definitions of sin θ and cos θ in terms of x, y, and r.