Find the value of x. Round your answer to the nearest tenth.
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
- 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: $$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $$
- 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: $$ \sin(37^\circ) = \frac{6}{\text{hypotenuse}} $$
- Calculate the hypotenuse
To find the hypotenuse, we can rearrange the equation: $$ \text{hypotenuse} = \frac{6}{\sin(37^\circ)} $$
- Find the value of $\sin(37^\circ)$
Using a calculator, we find: $$ \sin(37^\circ) \approx 0.6018 $$
- Substitute the sine value into the equation
Now, plug in the sine value: $$ \text{hypotenuse} \approx \frac{6}{0.6018} $$
- Calculate the hypotenuse
Perform the calculation: $$ \text{hypotenuse} \approx 9.966 $$
- Use the Pythagorean theorem to find x
Now we can find the adjacent side, (x), using the Pythagorean theorem: $$ x^2 + 6^2 = (9.966)^2 $$
- Calculate (x^2)
First, calculate ((9.966)^2): $$ (9.966)^2 \approx 99.33 $$ Then calculate (6^2 = 36).
Substitute into the Pythagorean theorem: $$ x^2 + 36 = 99.33 $$
- Solve for (x^2)
Rearranging gives: $$ x^2 = 99.33 - 36 $$
- 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.
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