Question

Solve for x and y. 30-60-90 triangles

Answer 1

Based on the given triangle, the value of y is 3√3 and the value of x is 3.

How to find the value of x and y

In a right triangle, use trigonometric ratios to solve for the lengths of the sides. In this case, we are given the hypotenuse (6) and the angle facing the opposite side (60 degrees). Let's solve for x and y.

Using the given information, determine the ratios for sine and cosine:

sin(60°) = opposite / hypotenuse

cos(60°) = adjacent / hypotenuse

sin(60°) = y / 6

cos(60°) = x / 6

To find the value of sin(60°), we know that sin(60°) = √3 / 2:

√3 / 2 = y / 6

Cross-multiplying:

√3 * 6 = 2 * y

6√3 = 2y

y = 3√3

Therefore, the value of y is 3√3.

To find the value of cos(60°), we know that cos(60°) = 1 / 2:

1 / 2 = x / 6

Cross-multiplying:

1 * 6 = 2 * x

6 = 2x

x = 3

Therefore, the value of x is 3.

Answer 2

• 
  `x = 3`

• 
  `y=3√(3)`

Explanation:

From the given right-angle triangle

The angle = ∠60°

• The adjacent to the angle ∠60° is x.

• The opposite to the angle ∠60° is y.

The hypotenuse = 6

Using the trigonometric ratio

cos 60° = adjacent / hypotenuse

substituting adjacent = x and hypotenuse = 6

cos 60° = x / 6

6 × cos 60° = x

x = 3

Thus, the value of x is:

• x = 3

And using the trigonometric ratio

sin 60° = opposite / hypotenuse

substituting opposite = y and hypotenuse = 6

`sin\:60^(\circ )\:=\:(y)/(6)\:`

`y=6\:*\:sin\:60^(\circ )`

`=6* (√(3))/(2)` ∵
`\sin \left(60^(\circ \:)\right)=(√(3))/(2)`

`=3√(3)`

Therefore, the value of y is:

• 
  `y=3√(3)`

Summary:

• 
  `x = 3`

• 
  `y=3√(3)`