10 TRIGONOMETRY Similarity Ratios Sketches Basic Reductions Special

10. TRIGONOMETRY • • Similarity Ratios Sketches Basic Reductions Special Triangles Equations 2 -D Problems 1

SIMILARITY n n If two triangles are similar, their corresponding angles are equal and their corresponding sides are in proportion. If then: ; ; and 2

Example: In the following triangles show that if then Similar Triangles 3

Summary: Similar Triangles n For any constant angle, the ratios for each similar triangle remain the same. n As the length of the hypotenuse increases, the length of the adjacent and opposite sides increase in the same proportion. 4

TRIG RATIOS Labelling Trigonometric 5

n The ratio is called the sine of the angle θ. n This can be written as sin θ = n The ratio n This can be written as cos θ = n The ratio n This can be written as tan θ = Definition of sinx is called the cosine of the angle θ Definition of cosx is called the tangent of the angle θ Introduction to Trig Functions Definition of tanx Trigonometric Ratios around the Origin 6

Example Consider the following diagram and then answer the questions that follow. Complete: (a) sin θ= θ= (c) tan θ = (b)cos (d)sin β= (e) cos β = (f) tan β = SOHCAHTOA 7

Exercise 1 1. (a) (b) (c) (d) (e) (f) Write down the following: sin C cos C tan C sin B cos B tan B 8

2. Write down the following: (a) (b) (c) (d) (e) (f) sin α cos α tan α Sin θ cos θ tan θ Trigonometric Ratios 9

Calculating the value of a trig function If you ensure that your calculator is on the DEG mode, you can evaluate trigonometric ratios 1. Evaluate the following rounded off to two decimal places where appropriate: (a) sin 57° (b) tan 67° (c) cos 24° (d) 3 sin 45° (e) 13 tan (45° + 54°) (f) (g) (h) (i) (j) 10

Calculating the size of an angle when given the trigonometric ratio n n Consider the equation sin β = 0, 5 Here we want to find the angle that gives the number 0, 5 In order to do this, we will make use of the button sin on the calculator If sin β = 0, 5, then we can find β by using the sequence: 2 nd. F sin 0, 5 = 11

Examples n Solve the following equations: (Round your answers off to two decimal places. ) (a) cos α = 0, 5 (c) tan α - 4, 123 = 0 (e) (b) 2 sin α = 1, 124 (d) cos 2 α = 0, 435 (f) (g) (h) (i) (j) (k) 12

Solving problems using trigonometric ratios Calculating the length of a side when given an angle and another side n Example 1: Calculate the length of AB in n You want side AB, which is opposite 36°. n You have side BC, the hypotenuse. You now need to create an equation involving the ratio and the angle 36°: sin 36° 13

Example 2 Calculate the length of BC to one decimal place. n n You want side BC, which is adjacent to 59°. You have side AB, which is opposite to 59°. You now need to create an equation involving the ratio and the angle 59°: 14

= tan 59º Using Trig Functions – The Ladder Using Trig Functions – The Stairs Using Trig Functions – The Skyscraper 15

EXERCISE 1. Calculate the length of PQ in 16

2. (a) Calculate the length of AB. (b) Calculate the length of BC. (c) What is the size of ? 17

3. By using the information provided on the diagram, calculate: (a) the length of AC. (b) the length of AB. 18

4. In the diagram, BD AC. Using the information provided, calculate the length of AC. 19

5. Using the information provided on the diagram , calculate the length of BC. Using Trig to Find the Perimeter Challenge! Find the Perimeter of an Irregular Shape 20

Calculating the size of an angle when given two sides n Example 4 We need to find angle θ. n We have side BC, which is adjacent to Side AC θ is the hypotenuse. n Finding an angle when given two sides 21

Therefore, we need to form an equation involving the ratio and the angle θ. 22

EXERCISE 1. (a) Calculate the size of θ. (b) Calculate the length of AC. 23

2. (a) Calculate the size of α. (b) Calculate the size of θ. 24

3. In , CD AB, A = , B = 40°, AD = l 5 cm and DB = l 6 cm. Calculate the size of θ 25

Reciprocals n e. g. 26

Exercise Calculate the following: a) b) c) d) e) 27

SKETCHES P(-3; 4) is a point on the Cartesian Plane and OP makes an angle of θ. n n Calculate the length of OP. OP² = (-3)² + (4)² = 25 OP = 5 units Calculate the value of: 4 -3 a) b) c) 28

EXERCISE K(-5; y) is a point on the Cartesian Plane and OK makes an angle of θ and equals 13. Determine the following: a) b) c) d) e) 29

BASIC REDUCTIONS “All Stations To Cape Town” CAST Diagram It is IMPORTANT to memorize the CAST diagram, as we can then determine the sign of the trigonometric function: • • • sinx: + Q 1 & Q 2 - Q 3 & Q 4 cosx: + Q 1 & Q 4 - Q 2 & Q 3 tanx: + Q 1 & Q 3 - Q 2 & Q 4 Quadrants of the Cartesian Plane 30

θ and 1800 - θ P` is the reflection of P about the y axis P`(-x; y) sin θ = y / r 1800 -θ P (x; y) sin(1800 - θ) = y/r O = sin θ cos θ = x/ r cos (1800 - θ ) = - x/r = - cos θ 31

θ and 1800 + θ P` is the reflection of P about the origin sin θ = y / r sin(1800 P (x; y) + θ) = - y/r 1800 + θ = - sin θ O cos θ = x/ r P`(-x; -y) cos (1800 + θ ) = - x/r = - cos θ 32

EXERCISE Simplify by means of reduction: a) b) c) d) e) f) 33

SPECIAL TRIANGLES 600 sin 600 = opp / hyp = cos 600 = adj / hyp = tan 600 = opp / adj = 2 600 1 Quiz: 60º Special Triangle 34

300 sin 300 = opp / hyp = cos 300 = adj / hyp = 300 2 tan 300 = opp / adj = 1 Quiz: 30º Special Triangle 35

450 n Determine the magnitude of the hypotenuse 1 cm = 450 1 cm 36

450 n Determine the following ratios sin 450 = opp / hyp = 1 cos 450 = adj / hyp = tan 450 = opp / adj = 450 1 1 Quiz: 45º Special Triangle 37

EXERCISE Calculate: a) b) c) d) e) f) 38

EQUATIONS Solving for y in trig equations 1. Find the value of y: a) y = sin 40º Calculator Work! Sine Function Box y = 0, 64 Cos Function Box b) y = cos 26º y = 0, 90 Tan Function Box c) y = tan 90º y = undefined Why? Think of the tan graph 39

Solving for the angle in trig equations 2. Solve for x: To findk a) sin x = ½ x = 30 ◦ b) cos x = 0, 78 x = 38. 74 ◦ c) tan x = 1 x = 45 ◦ To solve for the angle – type: SHIFT sin / cos / tan of the angle 40

EXERCISE Solve for x or y: a) sin x = 0. 37 b) y = 3 cos 90◦ c) y = tan 65◦ d) tan = 65◦ e) 5 sin x = 0. 5 f) 3 cos x = 1 41

2 -D Problems Angles of elevation and depression is the angle of elevation of C from A. is the angle of depression of A from C. 42

Example n n The angle of depression of a boat on the ocean from the top of a cliff is 55°. The boat is 70 meters from the foot of the cliff. 43

(a) What is the angle of elevation of the top of the cliff from the boat? (b) Calculate the height of the cliff. (a) The angle of elevation of the top of the cliff from the boat is 55°, i. e = 55°. (b) We can calculate the height of the cliff as follows: 44

Exercise 1. The Cape Town cable car takes tourists to the top of Table Mountain. The cable is 1, 2 kilometers in length and makes an angle of 40° with the ground. n Calculate the height (h) of the mountain. 45

2. An architectural design of the front of a house is given below. n The length of the house is to be 10 meters. n An exterior stairway leading to the roof is slanted to form an angle of elevation of 30° with ground level. The slanted part of the roof must be 7 meters in length. n 46

(a) Calculate the height of the vertical wall (DE). (b) Calculate the size of θ, the angle of elevation of the top of the roof (A) from the ceiling BCD. (c) Calculate the length of the beam AC. 47

3. In the 2006 Soccer World Cup, a player kicked the ball from a distance of 11 meters from the goalposts (4 meters high) in order to score a goal for his team. The distance traveled by the ball is in a straight line. n The angle formed by the pathway of the ball and the ground is represented by θ 48

(a) Calculate the largest angle θ for which the player will possibly score a goal. (b) Will the player score a goal if the angle θ is 22°? Explain. 49

4. Treasure hunters in a boat, at point A, detects a treasure chest at the bottom of the ocean (C) at an angle of depression of l 3° from the boat to the treasure chest. They then sail for 80 meters so that they are directly above the treasure chest at point B. In order to determine the amount of oxygen they will need when diving for the treasure, they must first calculate the depth of the treasure (BC). 50

Calculate the depth of the treasure for the treasure hunters. Colliding Ships Example 51