Question

Find the value of each variable. If your answer

is not an integer, write it in simplest radical
form with the denominator rationalized.

Answer 1

The values of x and y are 45√3 and 39 respectively.

What is Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse's length equals the sum of squares of the other two sides.

From the the figure

Tanθ = opposite/adjacent

Tan30 = 13√3/y

`(1)/( √(3) ) = (13 √(3) )/(y)`

y = 13*√3*√3

y = 13*3

= 39

Using Pythagorean theorem

`{x}^(2) = {y}^(2) + {(13 √(3)) }^(2)`

`{x}^(2) = {39}^(2) + {(13 √(3) )}^(2)`

x² = 1521 + 507

= 2028

`x = √(2028)`

x = 45√3

The values of x and y are 45√3 and 39 respectively.

Answer 2

x = 39/√2

y = (13/2)*√3

Explanation:

First let's write the only relation we need here:

Cos(θ) = (adjacent cathetus)/(hypotenuse)

or we also could use:

Sin(θ) = (opposite cathetus)/(hypotenuse).

For this problem, we know that the hypotenuse is H = 13*√3

Then if we steep on the 30° angle, the adjacent cathetus is x.

If we use the first relation we get:

cos(30°) = x/(13*√3)

With this, we can find the value of x.

We know that cos(30°) = (√3/√2)

Then:

(√3/√2) = x/(13*√3)

then:

(√3/√2)*(13*√3) = x = (√3*√3)*13/√2 = 3*13/√2 = 39/√2

x = 39/√2

Now if we use the angle of 60°, the adjacent side is y.

Then:

cos(60°) = y/(13*√3)

We know that:

cos(60°) = 1/2

Then:

(1/2) = y/(13*√3)

(1/2)*(13*√3) = y

(13/2)*√3 = y