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We are given that \(\sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}\) and we need to find the lengths of two triangles with a \(15^\circ\) angle.

To solve this, we can work with right triangles, since the sine function relates the opposite side to the hypotenuse in a right triangle. For a right triangle with an angle of \(15^\circ\):

\[
\sin(15^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

We are provided with the exact value of \(\sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}\). Let us assume the hypotenuse is 1 unit for simplicity. Then the opposite side can be calculated as:

\[
\text{opposite} = \sin(15^\circ) \times \text{hypotenuse} = \frac{\sqrt{6} - \sqrt{2}}{4} \times 1 = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

Now, using Pythagoras’ Theorem, we can calculate the adjacent side for this triangle:

\[
\text{adjacent} = \sqrt{\text{hypotenuse}^2 - \text{opposite}^2}
\]
\[
= \sqrt{1^2 - \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)^2}
\]

Let me calculate the adjacent side and finish solving this problem.For a right triangle with a \(15^\circ\) angle and a hypotenuse of 1 unit, the side lengths are:

- Opposite side: approximately \(0.259\)
- Adjacent side: approximately \(0.966\)

Since the problem asks for two triangles, the other triangle can be similar to the first, but scaled by a factor. For instance, if we scale the hypotenuse to 2 units, we can simply multiply the calculated side lengths by 2:

- For a hypotenuse of 2:
  - Opposite side: \(2 \times 0.259 = 0.518\)
  - Adjacent side: \(2 \times 0.966 = 1.932\)

Would you like further details or explanations?

Here are 5 related questions you can explore:
1. How do you calculate the sine of other angles using known values?
2. What is the exact value of \(\cos(15^\circ)\) and how can it be derived?
3. How does scaling affect the dimensions of a triangle with fixed angles?
4. How do you use trigonometric identities to verify values of sine or cosine?
5. How is the Pythagorean theorem applied to non-right triangles?

**Tip:** Knowing trigonometric identities for common angles like \(15^\circ\) can help in solving more complex trigonometry problems quickly!

Given sin(15°) = (√6 - √2) / 4, find the lengths of TWO triangles with a 15° angle.

Learn how to calculate the side lengths of two right triangles with a 15° angle, using the given sin(15°) = (√6 - √2) / 4. Apply the sine function and Pythagoras' theorem to solve this trigonometry problem.

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