###### O Level | IGCSE | Additional Mathematics

# Sketching of Trigonometric Curve

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## Trigonometric Curves

The sine, cosine and tangent graphs serve as an introduction to trigonometry, which is one of the most fundamental topics in mathematics. The graphs are a visual representation of trigonometry and show the relationship between sine, cosine and tangent functions. This relationship is used by trigonometric formulas to solve real-world problems.

### Sketching of Sine Curves

The sine curve is an example of a periodic function and it can be plotted on the complex plane or in the $xy$-plane. It is also called a sine wave or sinusoidal curve. The basic characteristic of this curve is that it repeats itself every $2\pi$ radians, or $360^\circ $. Sine curve can be used to determine velocity, acceleration, and displacement through the use of trigonometry.

**Practice 1**

The graph below shows the curve $y=\sin x$:

**Properties of $y=\sin x$**

Amplitude $=1$

Period, $T=2\pi $ or $360^\circ $

Max value of $\sin (x)=1$

Min value of $\sin (x)=-1$

**Properties of $y=a\sin (bx), a>0$**

So if we sketch a sine graph with increasing amplitude $(a=2)$ and $(a=3)$, it will look like this:

- $y=2\sin x$
- $y=3\sin x$

One way to control the shape of a sinusoidal wave is to alter its amplitude. A wave’s amplitude is a number measuring how high it gets, and what determines its maximum height is the maximum value of its $y$-coordinate. The next question is: what happens if we **increase** the value of $a$?Â

From the previous figure, we can see that the amplitude ($a$) of the sine curve is constant. When the amplitude of a sine curve **increases by one**, the height of the wave also **increases by one**. This is called vertical shift. Increasing the amplitude will make the sine curve steeper and we will end up with a graph twice as big.

The amplitude becomes $a$.

When we sketch a sine graph with variable $(b=2)$ and $(b=3)$, the graphs will look like this:

- $y=\sin 2x$
- $y=\sin 3x$

Another way to manipulate the shape of a sinusoidal wave is to change the period. The period of a trigonometry function is the extent to which the function runs through all possible values before reaching its original value and starting over again. What if we make the angle variable $(x)$ **bigger** by multiplying the angle variable with a number $(b)$?Â Â

The variable $b$ tells us the number of cycles between $0$ and $2\pi $. **Increasing the variable $b$** will result in a **higher frequency**, also known as the number of complete cycles, in the period.Â

In this case, the period is being reduced and hence, the formula for the period is $T=\frac{2\pi }{b}$ or $\frac{360^ \circ}{b}$.

To summarize, the properties of $y=a\sin (bx), a>0$ are

Amplitude $=a$

Period, $T=\frac{2\pi }{b}$ or $\frac{360^ \circ}{b}$

Max value of $\sin (bx)= 1$

Min value of $\sin (bx)= -1$

#### Question 1

Sketch the graph $y=5\sin3x$ for $0\le x\le 360^\circ$, stating the amplitude and period.

### Sketching of Cosine Curves

The graph below shows the cosine curve, $y=\cos x$.

**Properties of $y=\cos x$**

Amplitude $=1$

Period, $T=2\pi $ or $360^\circ $

Max value of $\sin (x)=1$

Min value of $\sin (x)=-1$

**Properties of $y=a\cos (bx),a>0$**

- $y=2\cos x$
- $y=3\cos x$

Similarly, the height of the curve corresponds directly to the amplitude of a cosine curve.

The figure on the left show the cosine graphs with amplitude $(a=2)$ and $(a=3)$.

- $y=\cos 2x$
- $y=\cos 3x$

Likewise, the frequency of the complete cycle in the period also increases correspondingly when we increase the variable $b$.

The figure on the left show the cosine graphs with $b=2$ and $b=3$.

To recap, the properties of $y=a\cos (bx),a>0$ are:

Amplitude $=a$

Period, $T=\frac{2\pi }{b}$ or $\frac{360^ \circ}{b}$

Max value of $\sin (bx)= 1$

Min value of $\sin (bx)= -1$

### Steps of Sketching Sine and Cosine Curve

- Decide the number of complete cycles required. No. of cycles$=\frac{\text{Given Domain}}{\text{Period}}$.
- Draw the grids out based on the number of complete cycles required. Take note that we need $4$ grids per cycle.
- Sketch the curve on the grid.
- Draw the $y$-axis.
- Mark down the Amplitude and the Period.
- Finally, draw the $x$-axis with respect to the line of equilibrium.

#### Question 2

Sketch the graph of $y=5\cos 3x$ for $0 \le x \le 240^\circ$, stating the amplitude and period.

### Sketching of Tangent Curves

From the sine and cosine curves, we know that the graph of a trigonometric function has an amplitude. The tangent curve, however, has no amplitude because the function is undefined. There is a line in the graph where the curve will never touch – the asymptote. Since the function approach to infinity at the asymptote, there is no maximum or minimum value.

The graph below shows the tangent curve, $y=\tan x$.

**Properties of $y=\tan x$**

Period, $T=\pi$ or $180^\circ $

Note: The period of the tangent curve is $\pi$ or $180^\circ $ which is different from that of the sine and cosine curve.Â

**Properties of $y=a\tan (bx),a>0$**

- $y=\tan 2x$
- $y=\tan 3x$

Period, $T=\frac{\pi}{b}$ or $\frac{180^\circ}{b}$

### Steps of Sketching Tangent Curve

- Decide the number of complete cycles required. No. of cycles$=\frac{\text{Given Domain}}{\text{Period}}$.
- Draw the grids out based on the number of complete cycles required. Take note that we need $2$ grids per cycle.
- Draw the $y$ and $x$ axes.
- Draw the asymptotes on grid lines $1,3,5,7,…$.
- Sketch the curve.
- Mark down the period and label the graph.

#### Question 3

Sketch the graph of $y=\tan (3x)$ for $0^\circ \le x \le 210^\circ$. State the period of the function.

### Sketching of Translated Curves

The sine and cosine curves can be translated in any of four directions: left, right, up, and down just like any other function. How do we identify the shift?

- If the addition or subtraction happens
**after**Â the sine and cosine functions, the function translatesÂ**vertically**. - If the addition or subtraction happens
**before**the sine and cosine functions, the function translatesÂ**horizontally**.

In this article, we will focus on the vertical translation:

- The function shifts upwards when addition happens after the function.
- The function shifts downwards when subtraction happens after the function.

- $y=\sin x$
- $y=2\sin x$
- $y=\sin 2x$

**Properties of $y=a\sin (bx)+c,a>0$:**

Amplitude $=a$

Period, $T=\frac{2\pi }{b}$ or $\frac{360^ \circ}{b}$

Max value of $a\sin (bx)+c= a+c$ when $a>0$

Min value of $a\sin (bx)+c= -a+c$ when $a>0$

- $y=\cos x+1$
- $y=2\cos x+1$
- $y=\cos 2x+1$

**Properties of $y=a\cos (bx)+c,a>0$:**

Amplitude $=a$

Period, $T=\frac{2\pi }{b}$ or $\frac{360^ \circ}{b}$

Max value of $a\cos (bx)+c= a+c$ when $a>0$

Min value of $a\cos (bx)+c=-a+c$ when $a>0$

#### Question 4

The equation of a curve is $y=2\sin 3x-1$ for $0 \le x \le 2\pi$.

(i)

Write down the maximum and minimum values of $y$.

(i) Write down the maximum and minimum values of $y$.

(ii)

State the amplitude, the period and the range of the function.

(ii) State the amplitude, the period and the range of the function.

(iii)

Hence, sketch the graph of $y$.

(iii) Hence, sketch the graph of $y$.

### Sketch of Trigonometric Functions with $a<0$

From the previous section, we know that we can alter the shape of the sinusoidal curve by manipulating the amplitude. The amplitude of a curve is the distance from the center to either peak or trough. It can be positive or negative. Now, what if the amplitude of the graph becomes **negative**?

We could expect a change in the amplitude of the curve. The new graph has a **negative** amplitude. It does not go up from left to right; it goes down from left to right! In fact, it looks just like our original graph except that it’s reflected across the $x$-axis!

$y=a\sin(bx),a<0$

$y=a\sin(bx)+c,a<0$

$y=a\cos(bx),a<0$

$y=a\cos(bx)+c,a<0$

$y=a\tan(bx),a<0$

#### Question 5

Sketch the graph of $y=-5\cos 2x+3$ for $0 \le x \le 3\pi$.

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