Question

The length of the hypotenuse of a right triangle is 20 feet. The tangent of one of the acute angles, angle 0, is 1.42. What is the length of the side opposite angle 0?

Answer 1

Oposite to the angle is 16.34

Explanation:

The tangent of angle O is 1.42

To get the angles we find tan^-1 1.42

Using scientific calculator or the table it is 54.8 degrees

Now we have the angle 54.8 degrees

The length of hypotenuse is 20 feet

To find the opposite

We use the sine Rule

Sin theta = opposite/hypotenuse

Sin 54.8 = opposite/hypotenuse

0.81714489833 = opposite/20

Opposite = 20 x 0.81714489833

= 16.3428979667 = 16.34 approximately 2 decimal places

The opposite side to the angle 54.8 is 16.34

Answer 2

16.4 feet.

Explanation:

We have been given that the length of the hypotenuse of a right triangle is 20 feet. The tangent of one of the acute angles, angle
`\theta`, is 1.42. We are asked to find the length of the side opposite angle
`\theta`.

We can represent our given information in an equation as:

`\text{tan}(\theta)=1.42`

Now, we will use arctan to solve for theta as:

`\theta=\text{tan}^(-1)(1.42)`

`\theta=54.85^(\circ)`

Now, we will use sine to solve for opposite side as sine relates opposite side of right triangle with hypotenuse.

`\text{sin}=\frac{\text{Opposite}}{\text{hypotenuse}}`

`\text{sin}(54.85^(\circ))=\frac{\text{Opposite}}{20}`

`20\cdot \text{sin}(54.85^(\circ))=\text{Opposite}`

`20\cdot 0.81764=\text{Opposite}`

`\text{Opposite}=20\cdot 0.81764`

`\text{Opposite}=16.3528`

`\text{Opposite}=16.4`

Therefore, the opposite side to angle theta is 16.4 feet.