MODULE - 11 TRIGONOMETRY

Understand that the similarity of triangles is fundamental to the trigonometric functions, sin theta, cos theta and tan theta, and is able to define and use the functions (LO 3 AS 5) Solve problems in two dimensions by using the trig functions in right – angled triangles and by constructing and interpreting geometric and trigonometric models. (LO 3 AS 6) Demonstrate the ability to work with various types of functions. (LO 2 AS 1 a)

Recognize relationships between variables in terms of numerical, graphical, verbal and symbolic representations. (LO 2 AS 1b) Generate as many graphs as possible, initially by means of point – point plotting, supported by available technology, to make and test conjectures and hence to generalize the effects of the parameters a and q on the graphs of the functions including:

Similar Triangles Before discussing the concepts of trigonometry in this module, we will need to briefly revise the concept of similar triangles. If two triangles are similar, their corresponding angles are equal and their corresponding sides are in proportion.

If then: (a) (b) Example: In the following triangles, show that if , then

Introduction to Trigonometry Trigonometry is the study of the relationships between lines and triangles. It has its origins in the study of astronomy where distances cannot be measured directly but have to be calculated. Evidence of such calculations has been found on ancient Babylonian clay tablets and in the relics of the ancient Egyptian civilization. Angles in Trigonometry are usually indicated by means of Greek letters: = theta, = beta, = alpha

Right-angled triangles are fundamental to the study of trigonometry.

The side AC, which lies opposite the right angles, is called the hypotenuse. The side AB lies opposite and side BC lies adjacent (next to) the angle .

Investigation 1 In the diagram which follows on the next page, (a) For each similar triangle, measure the length of the side opposite the 30 angle, adjacent to the 30° angle and the hypotenuse. Record your results in the table below. (b) Calculate the ratios in the table using a calculator. Round off to one decimal place if necessary.

Lengths of sides in mm Ratios Triangle Angle Opp side Adj. Hypotenuse

What can you conclude?

Investigation 2 In the diagram which follows on the next slide (a) For each similar triangle, measure the length of the side opposite the angle, adjacent to the 30°angle and the hypotenuse. Record your results in the table below. (b) Calculate the ratios in the table using a calculator. Round off to one decimal place if necessary.

Lengths of sides in mm Ratios Triangle Angle Opp side Adj Hypotenuse

Summary of conclusions from the two investigations: For any constant angle, the ratios for each similar triangle remain the same. As the length of the hypotenuse increases, the length of the adjacent and opposite sides increase in the same proportion.

Definitions of the trigonometric ratios

The ratio is called the sine of the angle. This can be written as sin = The ratio is called the cosine of the angle This can be written as cos = The ratio is called the tangent of the angle . This can be written as tan = Remember: A trigonometric ratio is a numerical value and not an angle

Example Consider the following diagram and then answer the questions that follow.

Complete: (a) sin = (b) cos = (c) tan = (d) sin (e) cos (f) tan

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

2. Write down the following: sin cos tan (f) tan

Evaluating the value of trigonometric ratios If you ensure that your calculator is on the DEG mode, you can evaluate trigonometric ratios without having to draw triangles as done in the previous investigations. Consider the first investigation done in this topic. It is clear from that investigation that sin .

We can use a calculator to do this calculation for us. On your calculator, press the button “sin” and then “30” and then “= “. Guess what? You get 0, 5. Awesome, isn’t it! This means that we can work out trigonometric ratios for any angle.

Examples Use a calculator to evaluate the following trigonometric ratios rounded off to two decimal places where necessary: (a) cos 20° = (b) tan 10° = (c) sin 30° = (d) sin 47° = (e) cos l46° = (f) tan 235° = (g) 3 cos 20° = (h) (j) (l) What do you notice about (k) and (l)?

EXERCISE 2 1. Evaluate the following rounded off to two decimal places where appropriate: (a) sin 57° (b) tan 67° (c) cos l24° (d) cos 320° (e) 3 sin 45° (f) 7 tan 58° (g) sin 130° (h) 13 tan (45° + 54°) (i) 25 sin 225°

(j) (k) (l) (m)

2. If means the same as , use this idea to calculate the value of the following rounded off to three decimal places: (a) sin 309° (b) sin 56° + cos 56° (c) sin 162° + cos 162° (d) sin 46° + cos 65° (e) Cos 32° (f) sin 124° (g) tan 124° (h) tan l35° (i) 5 cos 25° What conclusion can you make from the above exercise?

Calculating the size of an angle when given the trigonometric ratio Consider the equation sin = 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 = 0, 5, then we can find by using the sequence: 2ndF sin 0,5 =

Some calculators use the button INV or SHIFT instead of 2nd F. You need to make sure that you know how to use your calculator to do this work.

Examples Solve the following equations: (Round your answers off to two decimal places when necessary.) (a) cos = 0, 5 (b) 2 sin = 1,124 (c) tan - 4,123 = 0 (d) cos 2 = 0,435

(e) (f)

EXERCISE 3 Solve the following equations: (Round your answers off to two decimal places when necessary). (a) (b) (c) (d) (e)

(f) (g) (h) (i) (j) (k) (l)

Solving problems using trigonometric ratios Type 1 (Calculating the length of a side when given an angle and another side) Example 1 Calculate the length of AB in You want side AB, which is opposite 36°. You have side BC, the hypotenuse.

You now need to create an equation involving the ratio and the angle 36°: sin 36°

Example 2 Calculate the length of BC to one decimal place. 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°:

= tan 59 degrees

EXERCISE 4 (Round answers off to one decimal place in this exercise) Calculate the length of PQ in

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

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

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

5. Using the information provided on the diagram , calculate the length of BC.

Type 2 Calculating the size of an angle when given two sides Example 4 Calculate the size of to one decimal place. We need to find angle . We have side BC, which is adjacent to Side AC is the hypotenuse.

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

EXERCISE 5 Round answers off to one decimal place in this exercise (a) Calculate the size of . (b) Calculate the length of AC.

2. (a) Calculate the size of . (b) Calculate the

3. In , CD AB, A = , B = 40°, AD = l5 cm and DB = l6 cm. Calculate the size of

4. In the diagram below , is right-angled at C. It is given that AC = 4 units, tan A = and (a) Determine the length of BC. (b) Determine the length of AB. (c) Calculate the size of B. In , QS PR, QS = h units, PQ = m units, QR = n units.

(a) Express sin P in terms of h and m. (b) Express sin R in terms of h and n. (c) Hence show that m sin P = n sin R. (d) Now use the result in (c) to calculate the size of P if it is given that m = 40cm, n = 30cm and R = 80.

Angles of elevation and depression

is the angle of elevation of C from A. is the angle of depression of A from C.

Example 5 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.

(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:

ASSESSMENT TASK ASSIGNMENT (Round answers off to one decimal place in this 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. Calculate the height (h) of the mountain.

2. An architectural design of the front of a house is given below. The length of the house is to be 10 meters. 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.

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

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. The angle formed by the pathway of the ball and the ground is represented by

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

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 l3° 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). Calculate the depth of the treasure for the treasure hunters.

TRIGONOMETRIC FUNCTIONS The basic function y = sin x x [0°; 360°] 0° 30° 45° 60° 120° 135° 150° 180° y = sin x 210° 225° 240° 270° 300° 315° 330° 360° y = sin x

Now answer the following questions based on the graph: (a) The maximum value, is ________________________ (b) The minimum value is _______________________________ (c) Range: ______________ (d) Definition: Amplitude: [Distance between the max and min value] (i. e it is always POSITIVE) The amplitude is_________________

(e) Draw the graph of y = sin x for the interval x [-720°; 720°]

The graph of y = sin x repeats its basic shape every________________ This repetition is referred to as the _________ We say that the____________________of y = sin x is__________________________

Complete the following table for each of the graphs: Amplitude Maximum Minimum Period y = sin x y = 2 sin x y =3 sin x y = - sin x y = - 2 sin x

2. Sketch following graphs for x [0°; 360°]: y = cos x; y = 2 cos x; y = 3 cos x ; y = - cos x ; y= - 2 cos x

(l) The graph repeats itself every___________ We say the ________ is _________ (m) Draw the graph of y = tan x for the interval [-720° ; 720°]

The graph of y = tan x repeats its basic shape every__________________________. This repetition is referred to as the ____________________. We say that the __________ of y = tan x is ______________.

AMPLITUDE SHIFTS Sketch the following graphs for x [0°; 360°] y = sin x ; y = 2sin x ; y = 3sin x ; y = - sin x ; y = - 2sin x

Complete the following table for each of the graphs: Amplitude Maximum Minimum Period y = sin x y = 2 sin x y =3 sin x y =- sin x y=- 2 sin x

2. Sketch following graphs for x [0°; 360°]: y = cos x ; y= 2 cos x ; y = 3 cos x ; y = - cos x ; y= - 2 cos x

Complete the following table for each of the graphs: Amplitude Maximum Minimum Period y = cos x y= 2 cos x y = 3 cos x y = - cos x y = - 2 scos x

We see that the value of a effects the_______________ of the graph. The amplitude will be_____________________

On the same set of axes sketch the graphs of and for x

Vertical shifts y = sin x + d ; y = cos x + d ; y = tan x + d

Sketch y = sin x ; y = sin x + 1 and y = sin x - 2 ; x [0°;360°]

Graph Amplitude Maximum Minimum Period y = sin x y = sin x + 1

Sketch y = cos x; y = cos x + 2 and y = cos x - l ; x [0° ; 360°]

Graph Amplitude Maximum Minimum Period y = cos x y = cos x + 2 y = cos x - 1

We see that d affects the vertical shift of the graph. Complete the following: If d > 0 the graph moves vertically _______________________. If d < 0 the graph moves vertically _______________________.

Now, on your own (you may complete a table as we have done before) 3. Sketch : y = tan x ; y = tan x + l ; y = tan x – 2 ; then explain in your own words the differences between the three graphs. Have fun!! See you in the next module