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How do you solve for x using the SOHCAHTOA Rules (Using the Sine Rule)

To solve for \( x \) using the SOHCAHTOA rules, we need to use the sine function because we have the angle and the length of the adjacent side. 

In this right triangle, we are given:
- An angle of \( 32^\circ \)
- The length of the adjacent side is \( 10 \) units
- We need to find the length of the hypotenuse, \( x \)

Using the sine rule (SOHCAHTOA), we know:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

Since we are looking for the hypotenuse and we have the adjacent side and the angle, we use:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

Rearranging to solve for the hypotenuse (\( x \)):
\[ x = \frac{\text{adjacent}}{\cos(\theta)} \]

Substituting the given values:
\[ x = \frac{10}{\cos(32^\circ)} \]

Now, calculate \( \cos(32^\circ) \):
\[ \cos(32^\circ) \approx 0.848 \]

Substitute this value into the equation:
\[ x = \frac{10}{0.848} \approx 11.79 \]

Therefore, the length of the hypotenuse \( x \) is approximately \( 11.79 \) units.

Would you like more details or have any questions about this solution?

Here are 5 more questions you might find interesting:
1. How do you use the tangent function in a right triangle?
2. How do you find the missing side in a right triangle using the Pythagorean theorem?
3. What are the definitions of sine, cosine, and tangent?
4. How do you solve for an angle in a right triangle given two sides?
5. What is the difference between SOHCAHTOA and the Law of Sines?

**Tip:** When using trigonometric functions, always double-check which function (sine, cosine, or tangent) is appropriate based on the sides and angles given in the problem.

Learn how to solve for x using SOHCAHTOA rules (sine rule) with a step-by-step solution. This problem involves finding the length of the hypotenuse in a right triangle using trigonometric functions. Suitable for high school students studying trigonometry.

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