Find the length of side x in simplest radical form with a rational denominator.
60 degrees, 30 degrees. Right triangle
Side C (hypotenuse): not specified.
Side B: 4
Side A: X
To solve for side x, we can utilize the trigonometric ratios of a right triangle. In this case, we have a right triangle with angles measuring 60 degrees, 30 degrees, and 90 degrees.
Let's define side A as the opposite side to the 60-degree angle, side B as the opposite side to the 30-degree angle, and side C as the hypotenuse.
Using the trigonometric ratio for the sine function, we have:
sin(60°) = opposite / hypotenuse
sin(60°) = A / C
Re-arranging the equation to solve for side A, we get:
A = C * sin(60°)
Now, let's evaluate the sine of 60 degrees:
sin(60°) = √3 / 2
Substituting this value back into the equation, we have:
A = C * (√3 / 2)
Next, we can consider side B, which is given as 4. Since side B is opposite to the 30-degree angle, we can directly assign it as:
B = 4
Finally, we can use the Pythagorean theorem to relate sides A, B, and C in a right triangle:
A^2 + B^2 = C^2
Substituting the known values, we get:
(√3 / 2 * C)^2 + 4^2 = C^2
Simplifying the equation further:
(3 / 4) * C^2 + 16 = C^2
Multiplying both sides of the equation by 4 to eliminate the fraction:
3C^2 + 64 = 4C^2
Rearranging the equation by moving all terms to one side:
C^2 - 3C^2 = 64
Simplifying further:
-C^2 = 64
Dividing both sides by -1:
C^2 = -64
Since the length of any side of a triangle cannot be negative, this equation has no real solutions. Therefore, it is not possible to determine the length of side x in this case.
To find the length of side x in simplest radical form with a rational denominator, we can use the trigonometric functions sine and cosine.
Given that the right triangle has angles of 60 degrees and 30 degrees, we can use the relationships between the angles and the sides of the triangle. In this case, we have:
- The side opposite the 30-degree angle is x.
- The side opposite the 60-degree angle is 4.
We can use the sine function to determine the relationship between the side lengths and the angles:
sin(angle) = opposite / hypotenuse
For the 30-degree angle:
sin(30 degrees) = x / hypotenuse
For the 60-degree angle:
sin(60 degrees) = 4 / hypotenuse
To find the value of x, we need to isolate it in one of the equations. Let's start with the equation involving the 30-degree angle:
sin(30 degrees) = x / hypotenuse
To simplify, we need to find the value of sin(30 degrees). The sine of 30 degrees is equal to 1/2:
1/2 = x / hypotenuse
To solve for x, let's multiply both sides of the equation by the hypotenuse:
(1/2) * hypotenuse = x
Now we have x in terms of the hypotenuse.
Next, let's focus on the equation involving the 60-degree angle:
sin(60 degrees) = 4 / hypotenuse
To simplify, we need to find the value of sin(60 degrees). The sine of 60 degrees is equal to sqrt(3)/2:
sqrt(3)/2 = 4 / hypotenuse
To solve for the hypotenuse, we can cross-multiply:
sqrt(3) * hypotenuse = 2 * 4
Finally, we can solve for the hypotenuse:
hypotenuse = (2 * 4) / sqrt(3)
Now that we know the value of the hypotenuse, we can substitute it back into the equation involving x:
(1/2) * [(2 * 4) / sqrt(3)] = x
Simplifying the expression gives us:
4 / sqrt(3) = x
Therefore, the length of side x in simplest radical form with a rational denominator is 4 / sqrt(3).