Explanation
In trigonometry, we often use the values of:
- Sine (\(\sin\))
- Cosine (\(\cos\))
- Tangent (\(\tan\))
for the angles:
\(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\)
Instead of memorizing them, we can use a simple trick to find them quickly.
Formula / Rule
Step 1 – Sine values (\(\sin\))
Write:
\(\sqrt\frac{0}{4}, \sqrt\frac{1}{4}, \sqrt\frac{2}{4}, \sqrt\frac{3}{4}, \sqrt\frac{4}{4}\)
So:
\(\sin(0^\circ) = \sqrt\frac{0}{4} = 0\)
\(\sin(30^\circ) = \sqrt\frac{1}{4} = \frac{1}{2}\)
\(\sin(45^\circ) = \sqrt\frac{2}{4} = \frac{1}{\sqrt 2}\)
\(\sin(60^\circ) = \sqrt\frac{3}{4} = \frac{\sqrt 3}{2}\)
\(\sin(90^\circ) = \sqrt\frac{4}{4} = 1\)
Step 2 – Cosine values (\(\cos\))
Reverse the sine values:
\(\cos(0^\circ) = 1\)
\(\cos(30^\circ) = \frac{\sqrt 3}{2}\)
\(\cos(45^\circ) = \frac{1}{\sqrt 2}\)
\(\cos(60^\circ) = \frac{1}{2}\)
\(\cos(90^\circ) = 0\)
Step 3 – Tangent values (\(tan\))
Use:
\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
Example
Find \(tan(60^\circ)\).
Step 1:
\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
Step 2:
\(\tan(60^\circ) = \frac{\sqrt{3}}{2} ÷ \frac{1}{2}\)
Step 3:
Multiply by reciprocal
\(= (\sqrt{3}/2) \times (2/1)\)
Step 4:
\(= \sqrt{3}\)
Final Answer:
\(\tan(60^\circ) = \sqrt{3}\)