Calculating tangent

The \tan (tangent function) is defined as ‘the ratio of the sine function to the cosine function’.

In other words:

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

…where \theta is any angle.

Calculate \tan(45^\degree).

• \tan(45^\degree) = \frac{\sin(45^\degree)}{\cos(45^\degree)}
• \sin(45^\degree) = \frac{\sqrt{2}}{2}
• \cos(45^\degree) = \frac{\sqrt{2}}{2}
• \tan(45^\degree) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}
• \tan(45^\degree) = 1
• Answer: \tan(45^\degree) = 1.

Calculate \tan(60^\degree).

• \tan(60^\degree) = \frac{\sin(60^\degree)}{\cos(60^\degree)}
• \sin(60^\degree) = \frac{\sqrt{3}}{2}
• \cos(60^\degree) = \frac{1}{2}
• \tan(60^\degree) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}
• \tan(60^\degree) = \frac{\sqrt{3}}{1}
• \tan(60^\degree) = \sqrt{3}
• Answer: \tan(60^\degree) = \sqrt{3}.

Calculate \tan(30^\degree).

• \tan(30^\degree) = \frac{\sin(30^\degree)}{\cos(30^\degree)}
• \sin(30^\degree) = \frac{1}{2}
• \cos(30^\degree) = \frac{\sqrt{3}}{2}
• \tan(30^\degree) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}
• \tan(30^\degree) = \\frac{1}{\sqrt{3}}
• Answer: \tan(30^\degree) = \frac{1}{\sqrt{3}}.