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HAZ BIEN EL EJERCICIO ES PARA UNA BIMESTRAL

The problem given is to solve the equation:

\[
5 \cdot \sin^2\left(\frac{\pi}{4}\right) + 8 \cdot \cos^2\left(\frac{\pi}{6}\right)
\]

Let's solve it step by step.

1. **Calculate \(\sin\left(\frac{\pi}{4}\right)\):**

\[
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]

So:

\[
\sin^2\left(\frac{\pi}{4}\right) = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}
\]

2. **Calculate \(\cos\left(\frac{\pi}{6}\right)\):**

\[
\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\]

So:

\[
\cos^2\left(\frac{\pi}{6}\right) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}
\]

3. **Substitute these values into the equation:**

\[
5 \cdot \frac{1}{2} + 8 \cdot \frac{3}{4}
\]

4. **Simplify:**

\[
5 \cdot \frac{1}{2} = \frac{5}{2} = 2.5
\]

\[
8 \cdot \frac{3}{4} = 6
\]

So the expression becomes:

\[
2.5 + 6 = 8.5
\]

The final answer is:

\[
\boxed{8.5}
\]

Would you like more details or have any questions?

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Here are five related questions to consider:

1. How do the sine and cosine functions relate to the unit circle?
2. What is the general formula for \(\sin^2 x + \cos^2 x\)?
3. Can you solve similar trigonometric identities for different angles?
4. What happens if we change the coefficients in the original equation?
5. How do these trigonometric values apply to real-world problems?

**Tip:** Understanding the basic trigonometric identities can greatly simplify solving more complex problems.

Resuelva la siguiente ecuación: 5·sen²(π/4) + 8·cos²(π/6)

Learn how to solve the trigonometric equation 5·sen²(π/4) + 8·cos²(π/6) using trigonometric identities and simplification techniques. This problem is ideal for students in Grades 9-12 and covers key concepts like the Pythagorean identity.

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