Using the double angle formular, find the
ralue of the following angles in a
rationalize form: (1) (11)

Ask by Ryan Valdez. in Nigeria

Mar 30,2025

Question

Using the double angle formular, find the ralue of the following angles in a rationalize form: (1) $$ \operatorname{Cos} 15 $$ (11) $$ \sin 75 $$

UpStudy AI · Accepted Answer

To find the values using the double‐angle (or equivalently, half‐angle) formula, observe that $$15^\circ$$ is half of $$30^\circ$$ and $$75^\circ = 90^\circ - 15^\circ$$, so in fact

$$
\sin 75 = \cos 15.
$$

We start with the half‐angle formula for cosine:

$$
\cos \theta = \sqrt{\frac{1+\cos(2\theta)}{2}}.
$$

Setting $$\theta = 15^\circ$$ we have $$2\theta = 30^\circ$$. Therefore,

$$
\cos 15 = \sqrt{\frac{1+\cos 30}{2}}.
$$

Since

$$
\cos 30 = \frac{\sqrt{3}}{2},
$$

substitute it in:

$$
\cos 15 = \sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{\frac{2+\sqrt{3}}{2}}{2}} = \sqrt{\frac{2+\sqrt{3}}{4}} = \frac{\sqrt{2+\sqrt{3}}}{2}.
$$

Thus, we have

$$
\cos 15 = \frac{\sqrt{2+\sqrt{3}}}{2} \quad \text{and} \quad \sin 75 = \cos 15 = \frac{\sqrt{2+\sqrt{3}}}{2}.
$$
$$ \cos 15^\circ = \frac{\sqrt{2 + \sqrt{3}}}{2} \quad \text{and} \quad \sin 75^\circ = \frac{\sqrt{2 + \sqrt{3}}}{2} $$

Using the double angle formular, find the
ralue of the following angles in a
rationalize form: (1) (11)

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Using the double angle formular, find the ralue of the following angles in a rationalize form: (1) (11)