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1 Trigonometry Recall: hypotenuse opposite adjacent 1

2 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2

3 Example: Determine the value of x. a) b) c) 3

4 Radians The radian (rad) is a unit of angle measure. When a wheel completes a turn... 1) how many degrees has it moved? 2) how far has the rim of the wheel travelled? 3) How many times is the radius contained in the circumference? 4

5 r 1 rad r A B r When the distance covered on the circumference is equal to the length of the radius (i.e., ), the measure of the central angle is equal to 1 radian. There are radians in a circle. radians 5

6 We can convert between radians and degrees using the proportion or 6

7 Example: What is... a) in radians? b) rad in degrees? c) rad in degrees? d) in radians? 7

8 Work Book: Pages 194 & 195 Questions 1, 2 & 3 8

9 Arc Length Recall: central angle arc length 360 circumference Replacing with radians: Arc Length: L Note: must be in radians. 9

10 Examples: 1. Determine the length of the arc, given and. or 10

11 2. Determine the diameter of the circle if and. 11

12 3. Determine the measure of the central angle, if and. 12

13 4. Determine the length of the arc (using ) if and. or 13

14 Trigonometric Angles Terminal side y Initial side x A trigonometric angle: its vertex is at the origin. its initial side is on the x -axis (going right). its terminal side is obtained by a rotation. The angle is positive if the rotation is counterclockwise. The angle is negative if the rotation is clockwise, 14

15 Trigonometric angles are considered Coterminal, if they have the same terminal arm, but different rotations. y y x x 15

16 Determine the measure of the trigonometric angle shown below, in degrees and radians. y x 0 16

17 Represent the trigonometric angle rad. y x 17

18 In which quadrant would you find the following trigonometric angles? a) d) b) e) c) 18

19 Work B00k: p 196 # 7, 9, 10, 11 & 12 p 198 # 2 & 3 19

20 Trigonometric Circle The trigonometric circle is a unit circle centred at the origin. y Any point on the circle is called a trigonometric point x 1 Every trigonometric angle has a corresponding trigonometric point. 20

21 Known trig points: y 1 x

22 y x 1 Notice: and 22

23 and y 1 x The trig point for any trigonometric angle is. 23

24 Example: a) P(50 ) = b) 24

25 There are other "special angles" for which we can determine the exact coordinates of their trigonometric points. y x 1 25

26 1 The side opposite a 30 angle in a right triangle is equal to half the or rad hypotenuse. 26

27 2 27

28 3 28

29 y 1 x By reflecting these points around the axes, we can determine 12 more points with exact coordinates. 29

30 30

31 This circle with all of the exact trigonometric points is known as the trigonometric circle. 31

32 Example: Determine the exact values for the following points. a) b) 32

33 c) d) 33

34 e) f) g) 34

35 If... and, then... =? 35

36 Find the exact value of a) b) 36

37 Example: Determine the exact values of... a) 37

38 b) 38

39 c) 39

40 d) 40

41 Determine the sign of, and in each of the four quadrants. y x 41

42 Determine the missing value (exact) of a trigonometric point in quadrant 2, if. 42

43 Sum or Difference of Two Angles There are formulae that allow us to determine the sine, cosine and tangent of the sum or difference of two angles. Sine Cosine Tangent Sum Difference 43

44 We can use these rules to determine the exact values of some angles not found on the trig circle. Examples: Determine the exact values of

45 2. 45

46 Determine the exact values of

47 4. 47

48 5. 48

49 We can also use these rules to prove some statements. Examples: Show that

50 2. 50

51 3. 51

52 Can you find an expression that corresponds to These expressions are also rules and are known as the double angle formulas. 52

Trigonometry.notebook. March 16, Trigonometry. hypotenuse opposite. Recall: adjacent

Trigonometry.notebook. March 16, Trigonometry. hypotenuse opposite. Recall: adjacent Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)

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