Trigonometry: Graphs of trig functions (Grade 10) *

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1 OpenStax-CNX module: m Trigonometry: Graphs of trig functions (Grade 10) * Free High School Science Texts Project This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License Graphs of Trigonometric Functions This section describes the graphs of trigonometric functions. 1.1 Graph of sinθ Graph of sinθ Complete the following table, using your calculator to calculate the values. Then plot the values with sinθ on the y-axis and θ on the x-axis. Round answers to 1 decimal place. θ sinθ θ sinθ Figure 1 Let us look back at our values for sinθ * Version 1.1: Aug 2, :07 am Table 1

2 OpenStax-CNX module: m θ sinθ Table 2 As you can see, the function sinθ has a value of 0 at θ = 0. Its value then smoothly increases until θ = 90 when its value is 1. We also know that it later decreases to 0 when θ = 180. Putting all this together we can start to picture the full extent of the sine graph. The sine graph is shown in Figure 2. Notice the wave shape, with each wave having a length of 360. We say the graph has a period of 360. The height of the wave above (or below) the x-axis is called the wave's amplitude. Thus the maximum amplitude of the sine-wave is 1, and its minimum amplitude is -1. Figure 2: The graph of sinθ. 1.2 Functions of the form y = asin (x) + q In the equation, y = asin (x) + q, a and q are constants and have dierent eects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 3 for the function f (θ) = 2sinθ + 3. Figure 3: Graph of f (θ) = 2sinθ Functions of the Form y = asin (θ) + q : 1. On the same set of axes, plot the following graphs: a. a (θ) = sinθ 2 b. b (θ) = sinθ 1 c. c (θ) = sinθ d. d (θ) = sinθ + 1 e. e (θ) = sinθ + 2 Use your results to deduce the eect of q. 2. On the same set of axes, plot the following graphs: a. f (θ) = 2 sinθ b. g (θ) = 1 sinθ

3 OpenStax-CNX module: m c. h (θ) = 0 sinθ d. j (θ) = 1 sinθ e. k (θ) = 2 sinθ Use your results to deduce the eect of a. You should have found that the value of a aects the height of the peaks of the graph. As the magnitude of a increases, the peaks get higher. As it decreases, the peaks get lower. q is called the vertical shift. If q = 2, then the whole sine graph shifts up 2 units. If q = 1, the whole sine graph shifts down 1 unit. These dierent properties are summarised in Table 3. q > 0 a > 0 Figure 4 q < 0 Figure 6 Table 3: Table summarising general shapes and positions of graphs of functions of the form y = asin (x) + q Domain and Range For f (θ) = asin (θ) + q, the domain is {θ : θ R} because there is no value of θ R for which f (θ) is undened. The range of f (θ) = asinθ + q depends on whether the value for a is positive or negative. We will consider these two cases separately.

4 OpenStax-CNX module: m If a > 0 we have: 1 sinθ 1 a asinθ a a + q asinθ + q a + q a + q f (θ) a + q This tells us that for all values of θ, f (θ) is always between a + q and a + q. Therefore if a > 0, the range of f (θ) = asinθ + q is {f (θ) : f (θ) [ a + q, a + q]}. Similarly, it can be shown that if a < 0, the range of f (θ) = asinθ + q is {f (θ) : f (θ) [a + q, a + q]}. This is left as an exercise. tip: The easiest way to nd the range is simply to look for the "bottom" and the "top" of the graph. (7) Intercepts The y-intercept, y int, of f (θ) = asin (x) + q is simply the value of f (θ) at θ = 0. y int = f (0 ) = asin (0 ) + q = a (0) + q = q (7) 1.3 Graph of cosθ Graph of cosθ : Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with cosθ on the y-axis and θ on the x-axis. θ cosθ θ cosθ Figure 8

5 OpenStax-CNX module: m Let us look back at our values for cosθ Table 4 θ cosθ Table 5 If you look carefully, you will notice that the cosine of an angle θ is the same as the sine of the angle 90 θ. Take for example, cos60 = 1 2 = sin30 = sin (90 60 ) (8) This tells us that in order to create the cosine graph, all we need to do is to shift the sine graph 90 to the left. The graph of cosθ is shown in Figure 9. As the cosine graph is simply a shifted sine graph, it will have the same period and amplitude as the sine graph. Figure 9: The graph of cosθ. 1.4 Functions of the form y = acos (x) + q In the equation, y = acos (x) + q, a and q are constants and have dierent eects on the graph of the function. The general shape of the graph of functions of this form is shown in Figure 10 for the function f (θ) = 2cosθ + 3. Figure 10: Graph of f (θ) = 2cosθ Functions of the Form y = acos (θ) + q : 1. On the same set of axes, plot the following graphs: a. a (θ) = cosθ 2 b. b (θ) = cosθ 1 c. c (θ) = cosθ d. d (θ) = cosθ + 1

6 OpenStax-CNX module: m e. e (θ) = cosθ + 2 Use your results to deduce the eect of q. 2. On the same set of axes, plot the following graphs: a. f (θ) = 2 cosθ b. g (θ) = 1 cosθ c. h (θ) = 0 cosθ d. j (θ) = 1 cosθ e. k (θ) = 2 cosθ Use your results to deduce the eect of a. You should have found that the value of a aects the amplitude of the cosine graph in the same way it did for the sine graph. You should have also found that the value of q shifts the cosine graph in the same way as it did the sine graph. These dierent properties are summarised in Table 6. q > 0 a > 0 Figure 11 q < 0 Figure 13 Table 6: Table summarising general shapes and positions of graphs of functions of the form y = acos (x) + q Domain and Range For f (θ) = acos (θ) + q, the domain is {θ : θ R} because there is no value of θ R for which f (θ) is undened.

7 OpenStax-CNX module: m It is easy to see that the range of f (θ) will be the same as the range of asin (θ) + q. This is because the maximum and minimum values of acos (θ) + q will be the same as the maximum and minimum values of asin (θ) + q Intercepts The y-intercept of f (θ) = acos (x) + q is calculated in the same way as for sine. y int = f (0 ) = acos (0 ) + q = a (1) + q = a + q (14) 1.5 Comparison of Graphs of sinθ and cosθ Figure 15: The graph of cosθ (solid-line) and the graph of sinθ (dashed-line). Notice that the two graphs look very similar. Both oscillate up and down around the x-axis as you move along the axis. The distances between the peaks of the two graphs is the same and is constant along each graph. The height of the peaks and the depths of the troughs are the same. The only dierence is that the sin graph is shifted a little to the right of the cos graph by 90. That means that if you shift the whole cos graph to the right by 90 it will overlap perfectly with the sin graph. You could also move the sin graph by 90 to the left and it would overlap perfectly with the cos graph. This means that: sinθ = cos (θ 90) (shift the cos graph to the right) and cosθ = sin (θ + 90) (shift the sin graph to the left) (15) 1.6 Graph of tanθ Graph of tanθ Complete the following table, using your calculator to calculate the values correct to 1 decimal place. Then plot the values with tanθ on the y-axis and θ on the x-axis.

8 OpenStax-CNX module: m θ tanθ θ tanθ Figure 16 Let us look back at our values for tanθ Table 7 θ tanθ Table 8 Now that we have graphs for sinθ and cosθ, there is an easy way to visualise the tangent graph. Let us look back at our denitions of sinθ and cosθ for a right-angled triangle. sinθ cosθ = opposite hypotenuse adjacent hypotenuse = opposite = tanθ (16) adjacent This is the rst of an important set of equations called trigonometric identities. An identity is an equation, which holds true for any value which is put into it. In this case we have shown that tanθ = sinθ (16) cosθ for any value of θ. So we know that for values of θ for which sinθ = 0, we must also have tanθ = 0. Also, if cosθ = 0 our value of tanθ is undened as we cannot divide by 0. The graph is shown in Figure 17. The dashed vertical lines are at the values of θ where tanθ is not dened. Figure 17: The graph of tanθ.

9 OpenStax-CNX module: m Functions of the form y = atan (x) + q In the gure below is an example of a function of the form y = atan (x) + q. Figure 18: The graph of 2tanθ Functions of the Form y = atan (θ) + q : 1. On the same set of axes, plot the following graphs: a. a (θ) = tanθ 2 b. b (θ) = tanθ 1 c. c (θ) = tanθ d. d (θ) = tanθ + 1 e. e (θ) = tanθ + 2 Use your results to deduce the eect of q. 2. On the same set of axes, plot the following graphs: a. f (θ) = 2 tanθ b. g (θ) = 1 tanθ c. h (θ) = 0 tanθ d. j (θ) = 1 tanθ e. k (θ) = 2 tanθ Use your results to deduce the eect of a. You should have found that the value of a aects the steepness of each of the branches. The larger the absolute magnitude of a, the quicker the branches approach their asymptotes, the values where they are not dened. Negative a values switch the direction of the branches. You should have also found that the value of q aects the vertical shift as for sinθ and cosθ. These dierent properties are summarised in Table 9.

10 OpenStax-CNX module: m q > 0 a > 0 Figure 19 q < 0 Figure 21 Table 9: Table summarising general shapes and positions of graphs of functions of the form y = atan (x) + q Domain and Range The domain of f (θ) = atan (θ) + q is all the values of θ such that cosθ is not equal to 0. We have already seen that when cosθ = 0, tanθ = sinθ cosθ is undened, as we have division by zero. We know that cosθ = 0 for all θ = n, where n is an integer. So the domain of f (θ) = atan (θ) + q is all values of θ, except the values θ = n. The range of f (θ) = atanθ + q is {f (θ) : f (θ) (, )} Intercepts The y-intercept, y int, of f (θ) = atan (x) + q is again simply the value of f (θ) at θ = 0. y int = f (0 ) = atan (0 ) + q = a (0) + q = q (22) Asymptotes As θ approaches 90, tanθ approaches innity. But as θ is undened at 90, θ can only approach 90, but never equal it. Thus the tanθ curve gets closer and closer to the line θ = 90, without ever touching it.

11 OpenStax-CNX module: m Thus the line θ = 90 is an asymptote of tanθ. tanθ also has asymptotes at θ = n, where n is an integer Graphs of Trigonometric Functions 1. Using your knowldge of the eects of a and q, sketch each of the following graphs, without using a table of values, for θ [0 ; 360 ] a. y = 2sinθ b. y = 4cosθ c. y = 2cosθ + 1 d. y = sinθ 3 e. y = tanθ 2 f. y = 2cosθ 1 Click here for the solution Give the equations of each of the following graphs: Figure 23 Figure 24 Figure 25 Click here for the solution. 2 The following presentation summarises what you have learnt in this chapter

12 OpenStax-CNX module: m This media object is a Flash object. Please view or download it at < Figure 26 2 Summary We can dene three trigonometric functions for right angled triangles: sine (sin), cosine (cos) and tangent (tan). Each of these functions have a reciprocal: cosecant (cosec), secant (sec) and cotangent (cot). We can use the principles of solving equations and the trigonometric functions to help us solve simple trigonometric equations. We can solve problems in two dimensions that involve right angled triangles. For some special angles, we can easily nd the values of sin, cos and tan. We can extend the denitions of the trigonometric functions to any angle. Trigonometry is used to help us solve problems in 2-dimensions, such as nding the height of a building. We can draw graphs for sin, cos and tan 3 End of Chapter Exercises 1. Calculate the unknown lengths Figure 27 Click here for the solution In the triangle P QR, P R = 20 cm, QR = 22 cm and P ^R Q = 30. The perpendicular line from P to QR intersects QR at X. Calculate a. the length XR, b. the length P X, and c. the angle Q ^P X Click here for the solution A ladder of length 15 m is resting against a wall, the base of the ladder is 5 m from the wall. Find the angle between the wall and the ladder? Click here for the solution

13 OpenStax-CNX module: m A ladder of length 25 m is resting against a wall, the ladder makes an angle 37 to the wall. Find the distance between the wall and the base of the ladder? Click here for the solution In the following triangle nd the angle A ^B C Figure 28 Click here for the solution In the following triangle nd the length of side CD Figure 29 Click here for the solution A (5; 0) and B (11; 4). Find the angle between the line through A and B and the x-axis. Click here for the solution C (0; 13) and D ( 12; 14). Find the angle between the line through C and D and the y-axis. Click here for the solution A 5 m ladder is placed 2 m from the wall. What is the angle the ladder makes with the wall? Click here for the solution Given the points: E(5;0), F(6;2) and G(8;-2), nd angle F ^E G. Click here for the solution An isosceles triangle has sides 9 cm, 9 cm and 2 cm. Find the size of the smallest angle of the triangle. Click here for the solution A right-angled triangle has hypotenuse 13 mm. Find the length of the other two sides if one of the angles of the triangle is 50. Click here for the solution One of the angles of a rhombus (rhombus - A four-sided polygon, each of whose sides is of equal length) with perimeter 20 cm is 30. a. Find the sides of the rhombus. b. Find the length of both diagonals. Click here for the solution

14 OpenStax-CNX module: m Captain Hook was sailing towards a lighthouse with a height of 10 m. a. If the top of the lighthouse is 30 m away, what is the angle of elevation of the boat to the nearest integer? b. If the boat moves another 7 m towards the lighthouse, what is the new angle of elevation of the boat to the nearest integer? Click here for the solution (Tricky) A triangle with angles 40, 40 and 100 has a perimeter of 20 cm. Find the length of each side of the triangle. Click here for the solution