Question

Use a half-angle formula to find the exact value of $\sin 15°$. Then use a difference formula to find the exact value of $\sin 15°$. Show that the answers found are the same.

Short answer

The exact value and difference formula of $\sin 15°$are the same, it can be proven by first finding the value using half-angle formula and then using the difference formula.

The required formula is$sin\frac{\theta }{2}=\pm \sqrt{\frac{1-\mathrm{co}s\theta }{2}}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\sin 15°$

We have to find the exact value of the given expression using Half-angle formula,

$sin\frac{\theta }{2}=\pm \sqrt{\frac{1-\mathrm{co}s\theta }{2}}$

As $15°$is in the first quadrant. Then,

localid="1646498510525" $$\begin{gathered} \sin 15°=\sin {\left(\frac{30}{2}\right)}^{\circ } \\ \sin 15°=\sqrt{\frac{1-\cos \left(30°\right)}{2}} \\ \sin 15°=\sqrt{\frac{1-{\displaystyle \frac{\sqrt{3}}{2}}}{2}} \\ \sin 15°=0.2588 \end{gathered}$$

Step 2. Use the difference formula.

Consider the given question,

$\sin 15°$

Using the difference formula,

$$\begin{gathered} \sin 15°=\sin \left(45°-30°\right) \\ \sin 15°=\sin 45°\cos 30°-\cos 45°\sin 30° \\ \sin 15°=\left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right)-\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) \\ \sin 15°=\frac{\sqrt{3}-1}{2\sqrt{2}} \\ \sin 15°=0.2588 \end{gathered}$$

Hence, we can see that both the values are exactly same.