Find the value of x. Round your answer to the nearest tenth.

Question image

Understand the Problem

The question is asking to find the length of side x in a right triangle where one angle is 37 degrees and the side opposite to this angle is 6 units long. We will solve this using trigonometric functions, specifically the sine function, since we have the opposite side and need to find the adjacent side.

Answer

The value of $x$ is approximately $8.0$.
Answer for screen readers

The value of ( x ) is approximately $8.0$.

Steps to Solve

  1. Identify the appropriate trigonometric function

Since we have a right triangle and we know the angle (37 degrees) and the side opposite this angle (6), we will use the sine function, which is defined as: $$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $$

  1. Set up the equation using sine

We need to express the sine of the angle in terms of the known values. Let's substitute our values into the equation: $$ \sin(37^\circ) = \frac{6}{\text{hypotenuse}} $$

  1. Calculate the hypotenuse

To find the hypotenuse, we can rearrange the equation: $$ \text{hypotenuse} = \frac{6}{\sin(37^\circ)} $$

  1. Find the value of $\sin(37^\circ)$

Using a calculator, we find: $$ \sin(37^\circ) \approx 0.6018 $$

  1. Substitute the sine value into the equation

Now, plug in the sine value: $$ \text{hypotenuse} \approx \frac{6}{0.6018} $$

  1. Calculate the hypotenuse

Perform the calculation: $$ \text{hypotenuse} \approx 9.966 $$

  1. Use the Pythagorean theorem to find x

Now we can find the adjacent side, (x), using the Pythagorean theorem: $$ x^2 + 6^2 = (9.966)^2 $$

  1. Calculate (x^2)

First, calculate ((9.966)^2): $$ (9.966)^2 \approx 99.33 $$ Then calculate (6^2 = 36).

Substitute into the Pythagorean theorem: $$ x^2 + 36 = 99.33 $$

  1. Solve for (x^2)

Rearranging gives: $$ x^2 = 99.33 - 36 $$

  1. Calculate (x)

Now perform the subtraction: $$ x^2 = 63.33 $$

Finally, take the square root: $$ x \approx \sqrt{63.33} \approx 7.96 $$

Rounding to the nearest tenth: $$ x \approx 8.0 $$

The value of ( x ) is approximately $8.0$.

More Information

In a right triangle, the relationships between the angles and sides can be expressed using trigonometric functions. The sine function is particularly useful when the length of the opposite side and angle are known.

Tips

  • Not using the correct trigonometric function: Ensure you use sine for the opposite side and angle.
  • Incorrectly calculating the Pythagorean theorem: Double-check calculations and ensure you correctly rearrange the equation.
  • Forgetting to round to the nearest tenth: Always remember to apply the rounding instructions.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser