Sine and cosine functions

An angle can be either positive or negative.

Definition

Positive angle: is an angle measures in anticlockwise direction from the positive X-axis

Negative angle: is an angle measured in clockwise direction from the positive X-axis

Facts:

1. From the above figure if is a positive angle then the corresponding negative angle to is $(- \theta)$ or $(\theta - 360^\circ)$
2. If is a negative angle, its corresponding positive angle is $(\theta + 360^\circ)$

Example 1

Find the corresponding negative angle to the angle $\theta$ if:

1. $\theta = 58^\circ$
2. $\theta = 245^\circ$

Example 2

What is the positive angle corresponding to $-46^\circ$?

Solution

Since $\theta$ is negative, its corresponding positive angle is $\theta + 360^\circ$.

So $-46^\circ + 360^\circ = 314^\circ$

$\therefore -46^\circ$ corresponds to $314^\circ$

The angles included in this group are $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.

Because the angles $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ lie on the axes, their trigonometrical ratios are summarized in the following table.

Angle	$0^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
Sine	$0$	$1$	$0$	$-1$	$0$
Cosine	$1$	$0$	$-1$	$0$	$1$
Tangent	$0$	$\infty$	$0$	$\infty$	$0$

The $\triangle ABC$ is an equilateral triangle of side $2$ units.

For the angles $30^\circ$ and $60^\circ$ consider the following figures.

From the figure: $\cos 30^\circ = \frac{\sqrt{3}}{2}$ $\sin 30^\circ = \frac{1}{2}$ $\cos 60^\circ = \frac{1}{2}$ $\sin 60^\circ = \frac{\sqrt{3}}{2}$ $\tan 30^\circ = \frac{\sqrt{3}}{3} \quad \text{and} \quad \tan 60^\circ = \sqrt{3}$

For the angle $45^\circ$, consider the following triangle.

The following table summarizes the cosine, sine, and tangent of the angles $30^\circ$, $45^\circ$, and $60^\circ$.

NB: The following figure is helpful to remember the trigonometrical ratios of special angles from $0^\circ$ to $90^\circ$.

If we need the sines of the above given angles, for example, we only need to take the square root of the number below the given angle and then the result is divided by $2$.

Example 3

Find the sine, cosine, and tangents of each of the following angles:

1. $-135^\circ$
2. $120^\circ$
3. $330^\circ$

Example 4

Find the value of $\theta$ if $\cos \theta = -\frac{1}{2}$ and $0^\circ \leq \theta \leq 360^\circ$.

Solution

Since $\cos \theta$ is negative, then $\theta$ lies in either the second or third quadrants.

Now $\cos(180^\circ - \theta) = -\cos \theta$ or $\cos(180^\circ + \theta) = -\cos \theta$

So $-\frac{1}{2} = -\cos 60^\circ$

Thus $\theta = 180^\circ - 60^\circ = 120^\circ$ or $\theta = 180^\circ + 60^\circ = 240^\circ$

$\theta = 120^\circ$ or $\theta = 240^\circ$

Example 5

Consider below.

NB: $-\infty \leq \tan \theta \leq \infty$. The symbol $\infty$ means infinite.

Also you can observe that both $\sin \theta$ and $\cos \theta$ repeat themselves at the interval of $360^\circ$, which means $\sin \theta = \sin(\theta + 360^\circ) = \sin(\theta + 2 \times 360^\circ)$, etc.

And $\cos \theta = \cos(\theta + 360^\circ) = \cos(\theta + 2 \times 360^\circ)$.

Each of these functions is called a periodic function with a period $360^\circ$.

1. Using trigonometrical graphs in the interval $-360^\circ \leq \theta \leq 360^\circ$:
2. Find $\theta$ such that:
   • $\sin \theta = 0.4$
   • $\cos \theta = 0.9$

Solution

Example 6

Use the graph of $\sin \theta$ to find the value of $\theta$ if $4\sin \theta = -1.8$ and $-360^\circ \leq \theta \leq 360^\circ$.

Solution

$4\sin \theta = -1.8$

$\sin \theta = -1.8 \div 4 = -0.45$

$\sin \theta = -0.45$

So $\theta = -153^\circ$, $-27^\circ$, $207^\circ$, $333^\circ$

Example 7

Use the trigonometrical function graphs for sine and cosine to find the value of:

1. $\sin(-40^\circ)$
2. $\cos(-40^\circ)$

Solution

1. $\sin(-40^\circ) = -0.64$
2. $\cos(-40^\circ) = 0.76$