Answer:
Therefore, the value of
sin
3
(
1
0
∘
)
+
sin
3
(
5
0
∘
)
−
sin
3
(
7
0
∘
)
sin
3
(10
∘
)+sin
3
(50
∘
)−sin
3
(70
∘
) is approximately
−
0.3762
−0.3762.
Step-by-step explanation:
To find the value of \(\sin^3(10^\circ) + \sin^3(50^\circ) - \sin^3(70^\circ)\), we can use some trigonometric identities and relationships.
First, we use the fact that \(\sin(90^\circ - x) = \cos(x)\). This helps us relate the angles given:
\[
\sin(70^\circ) = \cos(20^\circ)
\]
Using the identity, \(\sin(50^\circ) = \cos(40^\circ)\).
Now, substituting these into our expression, we have:
\[
\sin^3(10^\circ) + \sin^3(50^\circ) - \sin^3(70^\circ) = \sin^3(10^\circ) + \cos^3(40^\circ) - \cos^3(20^\circ)
\]
Using \(\cos(40^\circ) = \sin(50^\circ)\) and \(\cos(20^\circ) = \sin(70^\circ)\), we get:
\[
= \sin^3(10^\circ) + \sin^3(50^\circ) - \sin^3(70^\circ)
\]
Notice that \(\sin(50^\circ) = \cos(40^\circ)\) and \(\cos(20^\circ) = \sin(70^\circ)\).
These simplifications don't straightforwardly resolve the problem, so we can instead try to find numerical values for these angles' sines and then compute:
- \(\sin(10^\circ) \approx 0.1736\)
- \(\sin(50^\circ) \approx 0.7660\)
- \(\sin(70^\circ) \approx 0.9397\)
Now, we calculate:
\[
\sin^3(10^\circ) \approx (0.1736)^3 = 0.0052
\]
\[
\sin^3(50^\circ) \approx (0.7660)^3 = 0.4492
\]
\[
\sin^3(70^\circ) \approx (0.9397)^3 = 0.8306
\]
Thus, the expression becomes:
\[
0.0052 + 0.4492 - 0.8306 = -0.3762
\]
Therefore, the value of \(\sin^3(10^\circ) + \sin^3(50^\circ) - \sin^3(70^\circ)\) is approximately \(-0.3762\).
