Q.Find the value of $\tan 22^\circ 30'$.

The key idea is to use the half-angle formula for tangent, since $22^\circ 30'$ is half of $45^\circ$.

Step 1: Recall the half-angle identity:

Step 2: Set $\theta = 45^\circ$. Then $\cos 45^\circ = \frac{\sqrt{2}}{2}$ and $\sin 45^\circ = \frac{\sqrt{2}}{2}$.

Step 3: Substitute: …

Why This Approach Works

The angle $22^\circ 30'$ is exactly half of $45^\circ$. When you see an angle like this — a common angle halved — the half-angle formulas for trigonometric functions are your natural tool. The tangent half-angle formula comes in several forms, but the most convenient one here is:

This form avoids square roots in the denominator and gives a clean algebraic result. Since $\theta = 45^\circ$ is a standard angle with known sine and cosine values, the calculation becomes straightforward.

Step-by-Step Solution

1. Identify the relationship.

We have $22^\circ 30' = \frac{45^\circ}{2}$. So let $\theta = 45^\circ$. Then $\tan 22^\circ 30' = \tan\frac{45^\circ}{2}$.

2. Apply the half-angle formula.

Using $\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta}$:

3. Substitute the known values.

We know $\cos 45^\circ = \frac{\sqrt{2}}{2}$ and $\sin 45^\circ = \frac{\sqrt{2}}{2}$.

4. Simplify the complex fraction.

Multiply numerator and denominator by 2:

5. Rationalise the denominator.

Multiply numerator and denominator by $\sqrt{2}$: …