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To solve for \( x \) in this triangle, we can use trigonometry. Here's the step-by-step solution:

1. **Identify the Known Angles and Side**:
   - Given angles: \( 47^\circ \) and \( 25^\circ \).
   - Hypotenuse (side opposite the right angle): \( 84 \).

2. **Determine the Remaining Angle**:
   Since the triangle contains a right angle (90°), the sum of the other two angles should be 90° as well. Therefore:
   \[
   \text{Remaining angle} = 90^\circ - 25^\circ = 65^\circ
   \]

3. **Apply the Sine Rule or Trigonometry**:
   To find \( x \), we can use the angle of \( 65^\circ \) opposite \( x \) and the known hypotenuse of \( 84 \).

   Using the **Cosine Rule** specifically:
   \[
   x = 84 \cdot \sin(65^\circ)
   \]

Let’s calculate this.The length of \( x \) is approximately \( 76.1 \) when rounded to one decimal place.

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would you solve for other sides of this triangle using trigonometric ratios?
2. What would change if the hypotenuse was a different length?
3. How could the Pythagorean theorem be applied in this context?
4. How can you determine the angles using inverse trigonometric functions if only the sides were known?
5. How does the sine function relate to the hypotenuse and opposite sides in any right triangle?

**Tip:** Remember that in right triangles, the sine function relates an angle to the ratio of the opposite side over the hypotenuse.

Find x. Round your answer to one decimal place.

Learn how to find the side length x in a right triangle with a given hypotenuse of 84 units and angles 47° and 25°. This solution demonstrates the use of trigonometric ratios, specifically the sine function, to solve for x. Suitable for students in Grades 9-11.

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