Question

Solve for x in the triangle. Round your answer to the nearest tenth.

14
x
68
X

Answer 1

x=15.1

Explanation:

Side definitions

In a right triangle, the side across from the right angle is always the hypotenuse.

The other two sides are termed "legs".

Once one of the other two angles is chosen as the angle to work from, then the two legs can be defined as the "opposite" and the "adjacent"

The leg touching the angle is the "adjacent" leg (or adjacent side), and the leg across from the angle is the "opposite" leg (or opposite side).

Trigonometric function definitions

`\sin(\theta)=\frac{\text{opposite side}}{\text{hypotenuse}}`

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

`\tan(\theta)=\frac{\text{opposite side}}{\text{adjacent side}}`

The given situation

In this situation, the given angle is in the bottom right. A value is given for the side opposite the given angle, and the requested value is on the hypotenuse.

So, the two sides of interest are the "opposite" and the "hypotenuse". The trigonometric function that relates these two sides is the sine function.

Calculating x

`\sin(\theta)=\frac{\text{opposite side}}{\text{hypotenuse}}`

`\sin(68^o)=((14))/((x))`

Evaluating the sine function in a calculator (in degree mode):

`0.9271838546...=(14)/(x)`

Multiply both sides by x:

`(0.9271838546...)x=\left ((14)/(x) \right)x`

`(0.9271838546...)x=14`

Divide both sides by the decimal number:

`\left (((0.9271838546...)x)/(0.9271838546...) \right)=(14)/(0.9271838546...)`

`x=15.0994864...`

Rounded to the nearest tenth, x=15.1