Derive function of `120^(@) ` from functions of `60^(@) and ` check by using relations between functions of supplementary angles.

Answer

To derive the function of \(120^\circ\) from the function of \(60^\circ\) and verify it using the relations between functions of supplementary angles, we can follow these steps:

### Step 1: Identify the angle
Let \(\theta = 60^\circ\). We want to find \(\sin(120^\circ)\).

### Step 2: Use the double angle formula
We can express \(120^\circ\) in terms of \(60^\circ\) using the double angle formula:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
Here, \(2\theta = 120^\circ\).

### Step 3: Substitute \(\theta\)
Substituting \(\theta = 60^\circ\) into the double angle formula gives:
\[ \sin(120^\circ) = \sin(2 \times 60^\circ) = 2 \sin(60^\circ) \cos(60^\circ) \]

### Step 4: Find the values of \(\sin(60^\circ)\) and \(\cos(60^\circ)\)
We know:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(60^\circ) = \frac{1}{2} \]

### Step 5: Substitute these values into the equation
Now substituting these values into the equation:
\[ \sin(120^\circ) = 2 \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{2}\right) \]

### Step 6: Simplify the expression
Simplifying the expression:
\[ \sin(120^\circ) = 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \]

### Step 7: Conclusion
Thus, we have derived that:
\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

### Verification using supplementary angles
To verify, we can use the property of supplementary angles:
\[ \sin(180^\circ - \theta) = \sin(\theta) \]
For \(\theta = 60^\circ\):
\[ \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) \]
Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), we have:
\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the derived function of \(120^\circ\) is:
\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]
And the verification confirms that our derived value is correct.

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