A right triangle is shown with an angle that measures 60 degrees. The leg adjacent to the 60 degree angle is labeled 5. The leg opposite of the 60 degree angle is labeled x. The hypotenuse is labeled y.
Find the value of x. If your answer is not an integer, leave it in simplest radical form. The diagram is not drawn to scale.
Using trigonometric ratios, we can solve for x:
sin(60°) = opposite/hypotenuse
sin(60°) = x/y
y = x/sin(60°)
cos(60°) = adjacent/hypotenuse
cos(60°) = 5/y
y = 5/cos(60°)
Since y is equal to y in both equations, we can set them equal to each other:
x/sin(60°) = 5/cos(60°)
x = 5(sin(60°)/cos(60°))
x = 5(tan(60°))
x = 5(sqrt(3))
Therefore, x equals 5√3.
Well, since we're dealing with a right triangle and one of the angles is 60 degrees, we can use the special triangle, the 30-60-90 triangle, as reference.
In a 30-60-90 triangle, the sides are in a ratio of 1:√3:2.
Since the leg adjacent to the 60-degree angle is labeled 5, we know that the ratio between that side and the hypotenuse (y) is 1:2. So, we can say that y = 2x.
Now, let's find x. Using the ratio 1:√3:2, we get 5/1 = x/√3. Cross-multiplying gives us 5√3 = x.
Therefore, the value of x is 5√3.
Keep in mind that the diagram is not drawn to scale, so the lengths might not be accurate representation. But hey, at least we found the value of x in simplest radical form!
To determine the value of x, we can use the trigonometric ratio for the sine function.
In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the hypotenuse.
So, in this case, sin(60 degrees) = x / y.
Since we know the length of the adjacent side (5) and the hypotenuse (y), we can use the Pythagorean theorem to find the value of y.
The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
So, in this case, 5^2 + x^2 = y^2.
Since the angle is 60 degrees, it can be used to form a 30-60-90 degree triangle.
In a 30-60-90 triangle, the ratio of the sides opposite the angles is always x, x√3, 2x.
Therefore, x is equal to the side opposite the 30-degree angle, which is 5 * √3.
Hence, the value of x is 5√3.
To find the value of x, we can use the trigonometric ratio of the sine function. In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
In this case, we have a right triangle with a 60 degree angle. The leg opposite the 60 degree angle is x, and the hypotenuse is y. Using the sine function, we can write:
sin(60 degrees) = x / y
Now, we need to find the value of sin(60 degrees). The sine of 60 degrees is equal to √3 / 2 (you can look this up in a trigonometric table or use a calculator).
So, we have:
√3 / 2 = x / y
To solve for x, we can multiply both sides of the equation by y:
(√3 / 2) * y = x
Finally, we substitute the given value of y with the value 5 and simplify:
x = (√3 / 2) * 5
x = 5√3 / 2
Therefore, the value of x is 5√3 / 2, which is the simplest radical form of the solution.