Derive an expression for $$ \sin(a+b) $$ using the sum formulas, where $$ a = 60^{\circ} $$ and $$ b = 45^{\circ} $$.

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UpStudy AI · Accepted Answer

To derive the expression for $$ \sin(a + b) $$ using the sum formulas where $$ a = 60^{\circ} $$ and $$ b = 45^{\circ} $$, follow these steps:

### Step 1: Use the Sine Sum Formula

The sine of the sum of two angles is given by the formula:

$$
\sin(a + b) = \sin a \cos b + \cos a \sin b
$$

### Step 2: Substitute the Given Angles

Substitute $$ a = 60^{\circ} $$ and $$ b = 45^{\circ} $$ into the formula:

$$
\sin(60^{\circ} + 45^{\circ}) = \sin 60^{\circ} \cos 45^{\circ} + \cos 60^{\circ} \sin 45^{\circ}
$$

### Step 3: Apply Known Values of Sine and Cosine

Recall the exact values for the sine and cosine of these standard angles:

$$
\begin{align*}
\sin 60^{\circ} &= \frac{\sqrt{3}}{2}, \
\cos 60^{\circ} &= \frac{1}{2}, \
\sin 45^{\circ} &= \frac{\sqrt{2}}{2}, \
\cos 45^{\circ} &= \frac{\sqrt{2}}{2}.
\end{align*}
$$

### Step 4: Substitute the Exact Values into the Expression

Plugging these values into the equation:

$$
\begin{align*}
\sin(105^{\circ}) &= \left( \frac{\sqrt{3}}{2} \right) \left( \frac{\sqrt{2}}{2} \right) + \left( \frac{1}{2} \right) \left( \frac{\sqrt{2}}{2} \right) \
&= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \
&= \frac{\sqrt{6} + \sqrt{2}}{4}.
\end{align*}
$$

### Final Result

Therefore, the expression for $$ \sin(60^{\circ} + 45^{\circ}) $$ is:

$$
\sin(105^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4}
$$
$$ \sin(105^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Derive an expression for using the sum formulas, where and .

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