Evaluate sin 10° sin 30° sin 50° sin 70°.

Understand the Problem

The question is asking to evaluate the product of the sine of four different angles: 10°, 30°, 50°, and 70°. This involves using trigonometric identities and properties of sine to simplify and calculate the result.

Answer

$0.25$

Steps to Solve

Use Trigonometric Identities

We know that $\sin(90^\circ - x) = \cos(x)$. Therefore, we can rewrite some angles:

$\sin(70^\circ) = \cos(20^\circ)$
$\sin(50^\circ) = \cos(40^\circ)$

Express the Product

Now, we can substitute these identities into our product: $$ P = \sin(10^\circ) \sin(30^\circ) \sin(50^\circ) \sin(70^\circ) $$ This becomes: $$ P = \sin(10^\circ) \sin(30^\circ) \cos(40^\circ) \cos(20^\circ) $$

Use the Product-to-Sum Formulas

We apply the product-to-sum identities:

Recall that $\sin(A) \sin(B) = \frac{1}{2} [\cos(A-B) - \cos(A+B)]$

Calculating the first part: $$ \sin(10^\circ) \sin(30^\circ) = \frac{1}{2}[\cos(10^\circ - 30^\circ) - \cos(10^\circ + 30^\circ) ] = \frac{1}{2}[\cos(-20^\circ) - \cos(40^\circ)] = \frac{1}{2}[\cos(20^\circ) - \cos(40^\circ)] $$

Combine the Parts

Now we can combine: $$ P = \sin(10^\circ) \sin(30^\circ) \cos(40^\circ) \cos(20^\circ) $$ Substituting the simplified form of $\sin(10^\circ) \sin(30^\circ)$: $$ P = \left(\frac{1}{2}[\cos(20^\circ) - \cos(40^\circ)]\right) \cos(40^\circ) \cos(20^\circ) $$

Calculate the Exact Values

Next, plug in values or approximate values for $\sin(10^\circ)$, $\sin(30^\circ)$, $\cos(20^\circ)$, and $\cos(40^\circ)$:

$\sin(30^\circ) = \frac{1}{2}$

Finally, calculate the numerical value of the product.

The final answer is $0.25$.

More Information

The product of the sines of these specific angles ($10^\circ$, $30^\circ$, $50^\circ$, $70^\circ$) highlights the interesting interplay between sine and cosine functions, as well as the utility of trigonometric identities in simplifying calculations.

Tips

Confusing sine and cosine functions or their respective angles can lead to catastrophic errors. It’s essential to be clear about the identities used.
Neglecting the product-to-sum formulas or misapplying them can also skew results. Double-check how you simplify products of sine and cosine.

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