Sure! Let's solve the expression step-by-step.
We are given the expression [tex]\(2 \frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}}\)[/tex].
First, let's break it down into the numerator and the denominator parts:
1. Calculate the numerator: [tex]\(\cos 40^{\circ} - \sin 30^{\circ}\)[/tex]
- The value of [tex]\(\cos 40^{\circ}\)[/tex] is approximately 0.766.
- The value of [tex]\(\sin 30^{\circ}\)[/tex] is 0.5.
So, the numerator is:
[tex]\[
\cos 40^{\circ} - \sin 30^{\circ} \approx 0.766 - 0.5 = 0.266
\][/tex]
2. Calculate the denominator: [tex]\(\sin 60^{\circ} - \cos 50^{\circ}\)[/tex]
- The value of [tex]\(\sin 60^{\circ}\)[/tex] is approximately 0.866.
- The value of [tex]\(\cos 50^{\circ}\)[/tex] is approximately 0.643.
So, the denominator is:
[tex]\[
\sin 60^{\circ} - \cos 50^{\circ} \approx 0.866 - 0.643 = 0.223
\][/tex]
3. Combine the numerator and the denominator in the expression
Now we substitute the values we have calculated into the expression:
[tex]\[
2 \frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}} = 2 \frac{0.266}{0.223}
\][/tex]
4. Divide the numerator by the denominator
Divide 0.266 by 0.223:
[tex]\[
\frac{0.266}{0.223} \approx 1.192
\][/tex]
5. Multiply by 2
Finally, multiply the result by 2:
[tex]\[
2 \times 1.192 \approx 2.384
\][/tex]
So, the value of the expression [tex]\(2 \frac{\cos 40^{\circ} - \sin 30^{\circ}}{\sin 60^{\circ} - \cos 50^{\circ}}\)[/tex] is approximately [tex]\(2.384\)[/tex].
