sin 47° + sin 61° + sin 11° + sin 25° is equal to

To find the sum of \( \sin 47^\circ + \sin 61^\circ + \sin 11^\circ + \sin 25^\circ \), we can utilize the sum-to-product identities.

First, let's group the sines in pairs where their sum can help us simplify the expression:

\[
\sin 47^\circ + \sin 61^\circ \quad \text{and} \quad \sin 11^\circ + \sin 25^\circ
\]

### Pair 1: \(\sin 47^\circ + \sin 61^\circ\)

Using the sum-to-product identities:

\[
\sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)
\]

For \( \sin 47^\circ + \sin 61^\circ \):

\[
\sin 47^\circ + \sin 61^\circ = 2 \sin \left( \frac{47^\circ + 61^\circ}{2} \right) \cos \left( \frac{47^\circ - 61^\circ}{2} \right)
\]

\[
= 2 \sin \left( \frac{108^\circ}{2} \right) \cos \left( \frac{-14^\circ}{2} \right)
\]

\[
= 2 \sin 54^\circ \cos (-7^\circ)
\]

Since \( \cos(-x) = \cos(x) \):

\[
= 2 \sin 54^\circ \cos 7^\circ
\]

### Pair 2: \(\sin 11^\circ + \sin 25^\circ\)

Similarly,

\[
\sin 11^\circ + \sin 25^\circ = 2 \sin \left( \frac{11^\circ + 25^\circ}{2} \right) \cos \left( \frac{11^\circ - 25^\circ}{2} \right)
\]

\[
= 2 \sin \left( \frac{36^\circ}{2} \right) \cos \left( \frac{-14^\circ}{2} \right)
\]

\[
= 2 \sin 18^\circ \cos 7^\circ
\]

### Adding both pairs together

Now, we add the two simplified expressions:

\[
\sin 47^\circ + \sin 61^\circ + \sin 11^\circ + \sin 25^\circ = 2 \sin 54^\circ \cos 7^\circ + 2 \sin 18^\circ \cos 7^\circ
\]

Factor out \(2 \cos 7^\circ\):

\[
= 2 \cos 7^\circ (\sin 54^\circ + \sin 18^\circ)
\]

### Simplifying the inner sine terms

Using known values for sine:
\[
\sin 54^\circ = \cos 36^\circ \quad \text{and} \quad \sin 18^\circ = \cos 72^\circ
\]

So,

\[
\sin 54^\circ + \sin 18^\circ = \cos 36^\circ + \cos 72^\circ
\]

It is known that:

\[
\cos 36^\circ = \frac{\sqrt{5} + 1}{4} \quad \text{and} \quad \cos 72^\circ = \frac{\sqrt{5} - 1}{4}
\]

So,

\[
\cos 36^\circ + \cos 72^\circ = \frac{\sqrt{5} + 1}{4} + \frac{\sqrt{5} - 1}{4} = \frac{2\sqrt{5}}{4} = \frac{\sqrt{5}}{2}
\]

Therefore,

\[
\sin 47^\circ + \sin 61^\circ + \sin 11^\circ + \sin 25^\circ = 2 \cos 7^\circ \left(\frac{\sqrt{5}}{2}\right) = \sqrt{5} \cos 7^\circ
\]

Since \(\cos 7^\circ\) is very close to 1 (given its small angle):

\[
\approx \sqrt{5}
\]

Thus,

\[
\sin 47^\circ + \sin 61^\circ + \sin 11^\circ + \sin 25^\circ \approx \sqrt{5}
\]

Would you like further details or have any questions? Here are 5 numbered questions for you to ask next:

1. Can you explain the sum-to-product identities in detail?
2. How do you simplify trigonometric expressions using identities?
3. What are the exact values of sine and cosine for special angles?
4. How do you derive the sum-to-product formulas?
5. Can you solve another trigonometric expression?

Tip: Familiarize yourself with trigonometric identities, as they can significantly simplify complex expressions.

Learn how to calculate the sum of sine values for angles 47°, 61°, 11°, and 25° using trigonometric identities. This problem involves applying the sum-to-product identities for sine and simplifying trigonometric expressions. Suitable for high school students studying trigonometry.

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