From the chapter of Trigonometry, we have learnt about the trigonometric functions involving sine, cosine and tangent.
They are real functions which relate an angle of a right-angled triangle to ratios of two side lengths, which aid us in finding unknown lengths and angles of a triangle and provide us with useful formulae like the sine and cosine rules.
Today, we will learn how to sketch the graphs of sine, cosine and tangent functions. Let’s begin!
In this note, you will learn:
· How to sketch Sine, Cosine and Tangent graphs and the different properties they possess
From the diagram above, you can see the different types of trigonometric function graphs.
There are a number of properties you must know and understand before you can draw/sketch a trigonometric graph:
Range: The set of possible output values, which are shown on the y-axis, for different values of x you input
Maximum/Minimum Values: The point where the graph has a vertex at the highest and lowest point respectively
Amplitude: The height from the center line to the maximum or minimum points of the graph
Period: The length of the smallest interval that contains exactly one copy of the repeating pattern of the graph
Now that we know what the different properties stands for, the table below shows the different values for different properties of the sine, cosine and tangent graphs.
*Note: Rotational symmetry of order 2 basically means that by rotating the graph, there will only be two positions in which the graph looks exactly the same.
The table above have shown the basic natural properties of sine, cosine and tangent graphs. However, questions involving trigonometric functions will not always present itself in this simple manner, hence we must learn the way the different properties work when there are coefficients that comes along with the trigonometric functions and its arguments.
Let’s take a look at how the different properties changes in the table below!
Let’s take a look at a few examples below:
1. The diagram below shows part of the graph of y = cos x and y = k, intersecting at three points where x = b, x = a and x = c. Find, in terms of a, the value of
i) b,
ii) c.
Solution:
i) Since the graph of y = cos x is symmetrical about the y-axis, then b = -a.
ii) Method 1:
Since the graph of y = cos x is symmetrical about the vertical line x = π, then the
mid-value of a and c is π,
i.e.
(a + c)/ 2 = π
a + c = 2 π
c = 2π – a
Method 2: Since the graph of y = cos x is symmetrical about the vertical line x = π. Then the horizontal distance from x = 0 to x = a is equal to the horizontal distance from x = c to x = 2π.
Therefore, c + a = 2π
c = 2π – a
2. The equation of a curve is y = 2 cos 3x + 2, where x is in degrees.
i) Write down the maximum and minimum values of y.
ii) State the range, the amplitude and the period of the function.
iii) Sketch the graph of y = 2 cos 3x + 2 for 0° ≤ x ≤ 360°.
Solution:
i) Maximum value of y = 2 x 1 + 2
= 4
Minimum value of y = 2 x (-1) + 2
= 0
ii) Range is 0 ≤ y ≤ 4
Amplitude = 2
Period = 360°/ 3 = 120°
iii)
3. The equation of a curve is y = 2tan 4x, where x is in radians.
i) State the period of the function.
ii) Hence, sketch the graph o y = 2tan 4x for 0 ≤ x ≤ 2π.
Solution:
i) Period = π/4
ii)
Note: The value 2 in front of the tangent function does not affect the shape of the graph, because a tangent graph does not have an amplitude since it does not have any maximum/minimum points!
Bonus question for you to practice!
4. The diagram shows the graph of y = a sin bx + c, where a, b, and c are integers and b > 0, for 0 ≤ x ≤ 2π. The graph has a maximum point at (3π/2, 3) and a minimum point at
(π/2, -1). Find the value of a, b and c.
And that’s all for today, students! Math Lobby hopes that after this article, you have a clear understanding on this topic of sine, cosine and tangent graphs and is hence equipped with the necessary skills to sketch graphs of different trigonometric functions!
Do note that this is only the first part of the article, and there will be a part two coming up next, so stay tuned to find out!
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