Sure! Let's solve the expression \(\sin \left(45^{\circ} - 30^{\circ}\right) = \sin 45^{\circ} \cdot \cos 30^{\circ} - \cos 45^{\circ} \cdot \sin 30^{\circ}\) step by step.
### Step 1: Understand the Expression
We need to verify the formula:
[tex]\[
\sin \left(45^{\circ} - 30^{\circ}\right) = \sin 45^{\circ} \cdot \cos 30^{\circ} - \cos 45^{\circ} \cdot \sin 30^{\circ}
\][/tex]
### Step 2: Calculate the Individual Trigonometric Values
#### For \( \sin 45^\circ \):
[tex]\[
\sin 45^\circ \approx 0.7071
\][/tex]
#### For \( \cos 30^\circ \):
[tex]\[
\cos 30^\circ \approx 0.8660
\][/tex]
#### For \( \cos 45^\circ \):
[tex]\[
\cos 45^\circ \approx 0.7071
\][/tex]
#### For \( \sin 30^\circ \):
[tex]\[
\sin 30^\circ \approx 0.5000
\][/tex]
### Step 3: Substitute the Values into the Expression
Substitute these values into the given equation:
[tex]\[
\sin 45^\circ \cdot \cos 30^\circ - \cos 45^\circ \cdot \sin 30^\circ
\][/tex]
Substitute the known values:
[tex]\[
0.7071 \cdot 0.8660 - 0.7071 \cdot 0.5000
\][/tex]
### Step 4: Perform the Multiplication
Calculate each product:
[tex]\[
0.7071 \cdot 0.8660 \approx 0.6124
\][/tex]
[tex]\[
0.7071 \cdot 0.5000 \approx 0.3536
\][/tex]
### Step 5: Subtract the Results
Subtract the second product from the first product:
[tex]\[
0.6124 - 0.3536 \approx 0.2588
\][/tex]
### Step 6: Verify the Result
Thus, we have:
[tex]\[
\sin \left(45^{\circ} - 30^{\circ} \right) = 0.2588
\][/tex]
After evaluating all the steps, the values calculated are:
[tex]\[
\sin 45^\circ \approx 0.7071, \quad \cos 30^\circ \approx 0.8660, \quad \cos 45^\circ \approx 0.7071, \quad \sin 30^\circ \approx 0.5000
\][/tex]
Finally, we conclude:
[tex]\[
\sin \left(45^{\circ} - 30^{\circ} \right) = \sin 45^{\circ} \cdot \cos 30^{\circ} - \cos 45^{\circ} \cdot \sin 30^{\circ} \approx 0.2588
\][/tex]
Therefore, the given trigonometric identity holds true.
