Measurement

In this section, we will learn how to draw and interpret graphs of trigonometric functions such as sine, cosine and tangent. These graphs help us understand how angles and lengths relate in real-life and mathematical situations.

We have already studied important topics in Year 1 and Year 2 that will help us here.

In Year 1, we worked with linear equations, functions and gradients of straight lines, which introduced us to plotting points and understanding graph shapes. In Year 2, we explored trigonometric ratios and vectors, which involved working with angles and direction. These are key ideas in trigonometric graphs.

Now, we will build on these skills to sketch trigonometric graphs and use them to solve problems. This will deepen our understanding of patterns, waves and movement in both mathematics and the real world.

KEY IDEAS

• Understanding Trigonometric Functions: Recognising sine, cosine and tangent as functions that relate angles to ratios in right-angled triangles.

• Graphing sine, cosine and tangent functions: Drawing and interpreting the basic shapes of their graphs over given intervals (e.g. 0°–360°).

• Amplitude, Period and Phase Shift: Exploring how changes in the graphs reflect variations in height (amplitude), horizontal length (period) and horizontal movement (phase shift).

• Real-life applications of trigonometric graphs: Connecting the wave-like behaviours to real-world contexts such as sound, tides and circular motion.

• Interpreting values from trigonometric graphs: Using the graphs to estimate function values, identify maximum/minimum points, intercepts and key features.

• Comparing trigonometric graphs: Understanding similarities and differences among sine, cosine and tangent graphs and how transformations affect their shapes.

A trigonometric graph provides a visual representation of how a trigonometric function behaves across a range of angle values. In these graphs:

• The horizontal axis (x-axis) displays the angle measurement, typically denoted as θ (theta).

• The vertical axis (y-axis) shows the corresponding output value of the trigonometric function at each angle.

In this section, we will focus solely on the graphs of trigonometric functions using angles measured in degrees, although they may also be expressed in radians.

Here are the key characteristics of the three primary trigonometric graphs.

1. Sine Graph (y = sin x) Domain: All real numbers Range: (−1, 1) (inclusive) Key Features: A smooth, continuous wave that starts at the origin (0,0) and oscillates between −1 and 1 and 1 with a period of 360° (2π radians)

2. Cosine Graph (y = cos x) Domain: All real numbers Range: (−1, 1) (inclusive) Key Features: Similar wave shape to sine but starts at (0,1), showing how it has phase-shifted by 90° from the sine function

3. Tangent Graph (y = tan x) Domain: All real numbers except where x = 90°, 270°, etc.

Range: All real numbers (unbounded) The Graph of y = Sinθ Take a close look at the scenario below Scenario In many parts of Ghana, especially in the villages, children enjoy swinging on ropes tied to strong tree branches. When you swing back and forth, your movement follows a smooth and regular pattern, you go up, come down, pass the middle, then go up again on the other side.

If someone were to measure your height from the ground as you swing over time, they would see your motion go up and down like a wave and that wave shape is exactly what the sine graph shows.

In this lesson, we will learn how to sketch this kind of graph and use it to solve problems that involve wave-like movements.

This shows a diagram of the sine graph for −360° ≤ θ ≤ 360°

Figure 4.1: Graph of y = sin for Properties of the graph of

1. The graph does not start at the origin, but it does pass through it.

2. The graph is continuous and repeats every 360.

3. The maximum value is +1.

4. The minimum value is −1.

5. The zeros/roots are where is an integer. The zeros = {…, -360, -180, 0, 180, 360, …..}

6. The domain is all real numbers. Domain = {θ:θ ∈ R}

7. The range is −1 ≤ θ ≤ 1.

8. The sine and cosine graphs have the same shape which is a smooth wave. The only difference is that the cosine graph is just a shifted version of the sine graph.

9. The sine graph passes through the origin (0,0), while the cosine graph starts at (0,1). If you move the sine graph a little to the left, it becomes the cosine graph.

So, based on what we saw with the sine graph, we can now easily understand the cosine graph.

Now look at the shape of the cosine graph and its properties.

The graph of y = Cosθ The diagram shows a graph of y = Cosθ for −360° ≤ θ ≤ 360°

Figure 4.2: Graph of y = Cosθ for −360° ≤ θ ≤ 360° Properties of the graph of y = Cosθ

1. The graph does NOT pass through the origin, unlike the Sine and the Tangent graphs

2. The graph is continuous and repeats every 360°.

3. The maximum value is +1.

4. The minimum value is −1.

5. The zeros/roots are where is an integer. Zeros = {…,-270, -90, 90, 270, …..}

6. The domain is all real numbers. Domain = {θ:θ ∈ R} The range is −1 ≤ θ ≤ 1.

Hopefully you can now understand the relationship between the sine and cosine graphs, as shown or described in Figure 4.3 below.

The graphs of y = sinθ and y = Cosθ

Figure 4.3: Graph of y = sinθ and y = Cosθ for −360° ≤ θ ≤ 360 Relationship between the sine and the cosine graph:

1. The graphs have the same shape.

2. If you shift any of the graphs horizontally, either to the left or right by 90°, you will get the other graph. This means Cos(90 − θ) = Sinθ. Likewise, Sin(90 − θ) = Cosθ.

So, we also conclude that if Cosx = Siny, it means x + y = 90° The graph of y=tanθ This graph has a unique shape compared with the sine and cosine graphs

Figure 4.4 shows a diagram of the tan graph for −360° ≤ θ ≤ 360°.

Figure 4.4: Graph of y = tanθ for −360° ≤ θ ≤360° Properties of the graph of y = tanθ

1. The graph passes through the origin.

2. The graph is NOT continuous and repeats every 180.

3. It has no maximum value (maximum value approaches ∞).

4. It has no minimum value (minimum value approaches −∞).).

5. The zeros/roots are where is an integer. Zeros = {…,−360, −180, 0, 180, 360,..}

6. Domain = {θ:θ ∈ R, except θ = 90° +180°n where n is an integer}

7. The range is -∞ ≤ θ ≤ ∞ or y∈R.

8. Vertical asymptotes at θ = 90° + 180°n where n is an integer. (Remember that an asymptote is a line that the graph will approach but never meet.)

Since the graphs of sine, cosin, and tangent have been explored, it is now time to learn how to draw trigonometric graphs. This lesson will focus only on the sine graph and the cosine graph Drawing Trigonometric Graphs Drawing a trigonometric graph follows the same procedure as drawing a quadratic graph

Example 4.1

a. Copy and complete the Table 4.1.

Table 4.1: Partially completed table for the relation y = 1.5 sinx x (degrees) 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° y = 1.5sinx 0 1.3 0 -1.5 0

b. Using a scale of 2 cm to 30° on the x-axis and 2 cm to 1 unit on the y-axis, draw the graph of y = 1.5sinx for 0° ≤ θ ≤ 360°

c. Use your graph to find:

i. the maximum point

ii. the minimum point

Solution:

Table 4.2: Completed table for the relation y = 1.5 sinx x (degrees) 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° y = 1.5sinx 0 0.75 1.3 1.5 1.3 0.75 0 −0.75 −1.5 −1.3 −0.75 0 For example:

When x = 30 = 1.5 sin30 = 0.75 When x = 60° y =1.5 sin60 = 1.3 Now we plot the above table values

Figure 4.5: Graph of y = 1.5sin for 0°≤x≤360

c. From the graph below:

i. The maximum point is (90°, 1.5)

Figure 4.6: A zoom in on the maximum part of the graph of y = 1.5sinx

iii. The minimum point is (270°, 1.5)

Figure 4.7: A zoom-in on the minimum part of the graph of y=1.5sinx

Example 4.2:

a. Copy and complete the Table 4.3.

Table 4.3: Partially completed table for the relation y = 2cosx x (degrees) 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° y = 2cosx 2 0 -1 -2 0 1 2

b. Using a scale of 2 cm to 30° on the and 2 cm to 1 unit on the , draw the graph of y = 2cosx for 0° ≤ θ ≤ 360°

c. Use your graph to find the minimum point

Solution:

Table 4.4: Completed table for the relation y = 2cosx x (degrees) 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° y = 2cosx 2 1.73 1 0 -1 -1.73 -2 -1.73 -1 0 1 1.73 2 When x = 30° y = 2cos (30) = 1.73 When x = 60° y =2cos (60) = 1 When x =150° y = 2cos (150) = −1.73 When x = 210° y=2cos (210) = −1.73 When x = 240° y = 2cos(240) = −1 When x = 300° y = 2cos(300) = 1 When x = 330° y = 2cos(330) = 1.73 Plot the points from the completed table

Figure 4.8: Graph of y = 2cosx for 0 ° ≤ x ≤ 360°

c. From the graph, the minimum point is (180°, −2)

Example 4.3:

Copy and complete the table of the function y = 3sin(x) 2cos(2x) for the interval 0° ≤ x ≤ 180°

Table 4.5: Partially completed table for the relation y = 3sin(x) − 2cos(2x) x (degrees) 0° 30° 60° 90° 120° 150° 180° y = 3sin(x) - 2cos(2x) 0.5 -2.0 Using a scale of 2 cm to 30° on the x-axis and 2 cm to 1 unit on the y-axis, draw the graph of y = 3sin(x) − 2cos(2x) for the interval 0° ≤ x ≤ 180°

Solution:

Table 4.6: Completed table for the relation y = 3sin(x) − 2cos(2x) x (degrees) 0° 30° 60° 90° 120° 150° 180° y = 3sin(x) - 2cos(2x) -2.0 0.5 3.60 5 3.60 0.5 -2.0 When x = 0° y = 3sin(0) − 2cos(2 × 0) = −2 When x = 60° y = 3sin(60) − 2cos(2 × 60) =3.60 When x = 90° y = 3sin(90) − 2cos(2 ×) = 5 When x = 120° y = 3sin(120) − 2 cos(2 × = 3.60 When x = 150° y = 3sin(150) − 2cos(2 × = 0.5 Plotting the points on the graph

Figure 4.9: Graph of y = 3sin x − 2cos2x for 0° ≤ θ ≤ 180°

Example 4.4:

a. Copy and complete the table of values for the relation y = 2 + 3sinθ

Table 4.7: Partially completed table for y = 2 + 3sinθ θ 0° 30° 90° 150° 180° 210° 270° 330° 360° y = 2 + 3sinθ 2 3.5 3.5 0.5 0.5 2

b. Graph Plotting Instructions Using: Scale: 2 cm = 30° on the θ-axis, 2 cm = 1 unit on the y-axis for the interval . Draw the graph of y = 2 + 3sinθ

c. Use your graph to find:

i. The minimum value of y.

ii. The coordinates where the minimum occurs.

iii. The maximum point

iv. The values of θ where y = 2, correct to 1 decimal place.

Solution:

Table 4.8: Completed table for the relation θ 0° 30° 90° 150° 180° 210° 270° 330° 360° y = 2 + 3sinθ 2 3.5 5 3.5 2 0.5 −1 0.5 2 When x = 90° y = 2 + 3sin90° = 5 When x = 180° y = 2+3sin180 = 2 When x = 270° y = 2 + 3sin270° = −1

Figure 4.10: Graph of y = 2 + 3sinθ for 0° ≤ θ ≤ 360°

c. From the graph above, the

i. Minimum value of y is −1

ii. Minimum point (270°, −1).

iii. Maximum point (90°, 5).

iv. θ = 0°, 180°, 360°, From the graph, these are the points where the curve intersects the line y = 2 (Indicated by the blue horizontal line on the graph).

Determining Equations Using Trigonometric

Graphs Use trigonometric graphs to determine equations and solve related problems Trigonometric graphs are useful tools for identifying and understanding the equations of periodic functions such as sine, cosine and tangent. By observing key features of these graphs — including amplitude, period, phase shift and vertical shift — we can determine their corresponding equations. Once obtained, these equations can then be applied to solve a variety of real-life and mathematical problems involving periodic behaviour.

Finding the solution set from the trigonometry graph The following examples will guide you in learning how to solve trigonometric equations using graphs.

Example 4.5:

a. Copy and complete the table below for the relation y = sinx − cos x for values of x from 0° to 180° at intervals of 30°.

Table 4.9: Partially completed table for the relation y = sinx − cos x.

x 0° 30° 60° 90° 120° 150° 180° y = sinx − cos x −1.00 −0.37 0.37 1.37

b. Using a scale of 2 cm to on the x-axis and 2 cm to 1 unit on the y-axis draw a graph of y.

c. Use your graph to estimate, in the given interval, the truth set of the equations.

i. sinx − cos x = 0

ii. sinx − cos x = 1

Solution

Table 4.10: Completed table of y = sinx − cos x.

x 0° 30° 60° 90° 120° 150° 180° y = sinx − cos x −1.00 −0.37 0.37 1.00 1.37 1.37 1.00 When x = 90° y = sin90° − cos90° = 1 When x = 120° y = sin120° − cos120° = 1.37 When x = 180° y = sin180° − cos180° = 1

b. Graph of y = sinx − cos x

Figure 4.11: Graph of the relation y = sinx − cos x c.

i. From the graph, the curve intersects the line y = 0 at x ≈ 45° Hence, the solution set of sinx − cos x = 0 is given by: {x ∶ x ≈ 45°}.

ii. From the graph, the curve, and the line y = 1 intersect at x = 90° and 180° Hence, the solution set of sinx − cos x = 1 is given by: {x ∶ x = 90° and 180°}.

Example 4.6:

a. Copy and complete the table below for the relation y = 3 cos x for values of x from 0° to 360° at intervals of 45°.

Table 4.11: Partially completed table for the relation y = 3 cos x x 0° 45° 90° 135° 180° 225° 270° 315° 360° y = 3 cos x 0 −3 0

b. Using a scale of 2 cm to on the x-axis and 2 cm to 1 unit on the y-axis, draw a graph of y.

c. Use your graph to estimate, in the given interval, the truth set of the equations

i. 2 + 3 cos x = 0

ii. the minimum point of y = 3 cos x.

iii. equation of the line of symmetry.

Solution:

Table 4.12: Completed table for the relation x 0° 45° 90° 135° 180° 225° 270° 315° 360° y = 3 cos x 3 2.12 0 −2.12 −3 −2.12 0 2.12 3 When x = 0° y = 3cos(0) = 3 When x = 45° y = 3cos(45) = 2.12 When x = 135° y = 3cos(135) = −2.12 When x = 225° y = 3cos(225) = −2.12 When x = 315° y = 3cos(315) = 2.12 When x = 360° y = 3cos(360) = 3

Figure 4.12: Graph of y = 3 cos x c.

i. Comparing the two equations y = 3 cos x and 2 + 3 cos x = 0, we obtained y = −2.

Draw the line y = −2 on the graph and find its truth set. That is the values of x for which the line and the curve intersect.

Figure 4.13: Graph of and line y = 3cos x and line y = −2 From the graph, the line y = −2 intersects the curve at x ≈132.02° and x ≈ 229.5° as shown in the graph above.

ii. The minimum point is the turning point of the graph.

Figure 4.14: Graph of y = 3cos x showing the minimum point From the graph, the minimum point as indicated is (180°, −3).

The y-coordinate of this point is called the minimum value, which is −3. The minimum value occurs when x = 180°.

iii. The line of symmetry divides the curve within the range given into equal halves.

Figure 4.15: Graph of y = 3cos x showing the equation of the line of symmetry From the graph, the line of symmetry is indicated by the broken line. It intersects the x-axis at 180°. Hence, the line of symmetry is x = 180°.

Activity 4.1 Comparing Sine and Cosine graphs Materials Needed

• Graph paper

• Pencil and ruler

• Calculator

• Coloured pens or markers (e.g. blue and red)

• Table of values below Instructions

Step 1: Complete the Table

Table 4.13

θ (degrees) 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° y = sinθ y = cosθ

Step 2: Draw the Graphs

• Draw a set of axes on your graph paper or in your graph book.

• Let the horizontal axis (x-axis) represent angles from 0° to 360°, using a scale like:1 cm = 30°

• Let the vertical axis (y-axis) represent values from −1 to 1, using: 1 cm = 0.5 unit Plot:

• y = sinθ using a blue pen

• y = cosθ using a red pen

Step 3: Reflect and Respond

On the same page (or at the back of your graph), answer these questions

1. Which graph starts at zero? Which one starts at one?

2. Do both graphs have the same height (amplitude)?

3. Do the graphs repeat after 360°?

4. How are the graphs similar?

5. How are they different?

6. If you were to draw a vertical line to divide each graph into two mirror parts, where would you draw it? Write the equation of the line of symmetry.

Learning Outcome: You will understand how sine and cosine graphs behave, and how they can be used to model real-life patterns that repeat.

1.

a. Copy and complete below for the relation y = 2sin x.

Table 4.14: Partially completed table for y = 2sin x.

x (degrees) 0° 45° 90° 135° 180° 225° 270° 315° 360° y = 2sin x 0 2 0 -2 0

b. Using a scale of 2 cm to 45° on the x-axis and 2 cm to 1 unit on the y-axis, draw the graph of y = 2sin x for 0°≤x≤360°.

c. Use your graph to find

i. The maximum point.

ii. The minimum point.

2.

a. Copy and complete the table below for the relation y = 3 cos x + 2 sin x for values of x from 0° to 180° at intervals of 30°

Table 4.15: Partially completed table of the relation y = 3 cos x + 2 sin x x 0° 30° 60° 90° 120° 150° 180° y = 3 cos x + 2sin x 3.00 3.23 -3.00

b. Using a scale of 2 cm to on the x-axis and 2 cm to 1 unit on the y-axis draw a graph of y = 3 cos x + 2 sin x.

c. Use your graph to estimate, in the given interval, the truth set of the equations. Correct your answer to one decimal place.

i. 6 cos x + 4 sin x = 1

ii. 3 cos x + 2 sin x = −1

3. The table below shows values of y = 2 + sinθ for 0° ≤ θ ≤ 360° for at intervals of 30°

Table 4.16

θ (degrees) 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360° y 2.00 2.50 2.87 3.00 2.87 2.50 2.00 1.50 1.13 1.00 1.13 1.50 2.00

a. Using a scale of 2 cm = 30° on the x-axis and 2 cm = 1 unit on the y-axis, plot the graph of y = 2 + sinθ for 0° ≤ θ ≤ 360°

b. From your graph, determine:

i. The maximum point.

ii. The minimum point.

c. Using your graph, find the truth set for each equation below, correct to one decimal place:

i. y = 2.5

ii. y = 1.5