Question

A right triangle is given. a right triangle is shown with the base labeled as 8.5 and the hypotenuse labeled as x. the angle theta is between the height and the hypotenuse and measures 40 degrees. determine the value for x.

a. 6.5
b. 7.1
c. 11.1
d. 13.2

Answer 1

To find the value of x, use the sine function in a right triangle. Plug in the values and calculate using a calculator. The answer is 13.2.

Step-by-step explanation:

To find the value of x, we can use the trigonometric functions in a right triangle. In this case, we have the angle theta as 40 degrees and the hypotenuse labeled as x. Using the sine function, we can write sin(theta) = opposite/hypotenuse. So, sin(40) = a/x. Rearranging the equation, we get x = a/sin(40). Now, we can plug in the values given. Since the base is labeled as 8.5, we have x = 8.5/sin(40). Using a calculator, sin(40) is approximately 0.643, so x = 8.5/0.643 = 13.2.

Answer 2

The value for the hypotenuse
`\( x \)` in the given right triangle is approximately 11.1. Therefore, option c is correct

In a right triangle with an angle
`\( \theta \)` and a base of length 8.5, we can find the length of the hypotenuse
`\( x \)` using the cosine function, which relates the angle of a right triangle to the adjacent side (base) and the hypotenuse.

The cosine function is defined as:

`\[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \]`

For the given triangle:

`\[ \cos(40^\circ) = (8.5)/(x) \]`

To find
`\( x \)`, we can rearrange the formula:

`\[ x = (8.5)/(\cos(40^\circ)) \]`

Using a calculator, we find the cosine of 40 degrees. Note that we must convert the angle from degrees to radians because the trigonometric functions in most programming languages and calculators expect the input in radians.

The conversion from degrees to radians is done using the formula:

`\[ \text{Radians} = \text{Degrees} * (\pi)/(180) \]`

So, for 40 degrees:

`\[ \text{Radians} = 40 * (\pi)/(180) \]`

Once we have the angle in radians, we can find the cosine of this angle and then calculate \( x \). Let's perform this calculation step by step.

The calculation for the hypotenuse \( x \) is as follows:

`\[ \text{Radians} = 40^\circ * (\pi)/(180) \]`

`\[ \cos(40^\circ) \approx 0.7660444431 \]`

`\[ x = \frac{\text{Base}}{\cos(\theta)} = (8.5)/(\cos(40^\circ)) \approx (8.5)/(0.7660444431) \]`

`\[ x \approx 11.095961959324368 \]`

The value for the hypotenuse
`\( x \)` in the given right triangle is approximately 11.1. Therefore, the correct answer is:

c. 11.1.