To find the value of the expression [tex]\(\sin 70^\circ - \sin 10^\circ - \sin 50^\circ\)[/tex], we will go through the following steps:
1. Determine the sine values of the angles involved:
- We need to find [tex]\(\sin 70^\circ\)[/tex], [tex]\(\sin 10^\circ\)[/tex], and [tex]\(\sin 50^\circ\)[/tex].
2. Values of sine at the specific angles:
- The value of [tex]\(\sin 70^\circ\)[/tex] is approximately [tex]\(0.9396926207859083\)[/tex].
- The value of [tex]\(\sin 10^\circ\)[/tex] is approximately [tex]\(0.17364817766693033\)[/tex].
- The value of [tex]\(\sin 50^\circ\)[/tex] is approximately [tex]\(0.766044443118978\)[/tex].
3. Substitute these values into the expression:
- Now, substitute the corresponding sine values into the expression [tex]\(\sin 70^\circ - \sin 10^\circ - \sin 50^\circ\)[/tex].
4. Perform the subtraction:
- First, calculate the result of [tex]\(\sin 70^\circ - \sin 10^\circ\)[/tex]:
[tex]\[
\sin 70^\circ - \sin 10^\circ \approx 0.9396926207859083 - 0.17364817766693033 = 0.766044443118978
\][/tex]
- Next, subtract [tex]\(\sin 50^\circ\)[/tex] from the result:
[tex]\[
0.766044443118978 - 0.766044443118978 = 0
\][/tex]
Therefore, the value of the expression [tex]\(\sin 70^\circ - \sin 10^\circ - \sin 50^\circ\)[/tex] is [tex]\(\boxed{0}\)[/tex].
