cos 0°

Definition

The value of cosine when the angle of the right angled triangle equals to zero angle is called $\cos 0^\circ$.

The value of cosine of angle $0^\circ$ can be evaluated by calculating the ratio of lengths of adjacent side to hypotenuse when the angle of the right angled triangle is zero degrees.

Proof

The value of $\cos 0^\circ$ can be exactly evaluated geometrically in two different methods but process of them is same.

1

Fundamental approach

Evaluate the ratio of length of the adjacent side to length of the hypotenuse.

$$\frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$$

Trigonometrically, the ratio of lengths of adjacent side to hypotenuse is called cosine of an angle but the angle of the right angled triangle here is $0^\circ$. Therefore, the ratio between them is denoted by $\cos 0^\circ$.

$$\therefore \,\, \cos 0^\circ = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$$

The lengths of both adjacent side and hypotenuse are equal because length of the opposite side is zero when the angle of the right angled triangle is zero degrees according to the properties of the right angled triangle.

$Length \, of \, Adjacent \, side =$ $Length \, of \, Hypotenuse$

$$\cos 0^\circ = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse} = \frac{d}{d}$$

$$\therefore \,\, \cos 0^\circ = 1$$

2

Practical approach

The value of $\cos 0^\circ$ can be evaluated by constructing a right angled triangle with zero degrees angle by using geometric tools.

1. Use ruler and draw a horizontal straight line of any length. Call the left side endpoint of the line as point $D$.
2. Use protractor and coincide its middle point with point $D$ and also coincide its right side baseline with horizontal line. Find $0^\circ$ by considering bottom scale and the horizontal line is exact zero degrees line surprisingly.
3. Now, take ruler and compass, and then set the distance from needle point to point of the pencil lead to any length (for example $8$ centimetres). Now, draw an on arc on horizontal line from $D$ on $0^\circ$ angle line (horizontal line in this case). The arc cuts the zero angle line at point $E$.
4. Draw a perpendicular line from point $E$ to horizontal line by using ruler to construct a right angled triangle but it is not possible in this case. So, assume the line from point $E$ to horizontal line is drawn and it intersects the horizontal line at point $F$.

Thus, a right angled triangle (denoted by $\Delta EDF$) is constructed geometrically. The angle of the triangle is zero angle. Calculate the ratio of lengths of adjacent side to hypotenuse to get the value of $\cos 0^\circ$.

$$\cos 0^\circ = \frac{DF}{DE}$$

$$\implies \cos 0^\circ = \frac{8}{8}$$

$\therefore \,\, \cos 0^\circ = 1$

Result

The two methods have proved that the value of cosine of angle $0^\circ$ is $1$.

$\therefore \,\, \cos 0^\circ = 1$

Representation

It is written in three different methods based on three angle measuring systems.

It is written in sexagesimal system as follows.

$\cos 0^° = 1$

It is also written in circular system as follows.

$\cos 0 = 1$

In centesimal system, it is also expressed as follows.

$\cos 0^g = 1$