in the special right triangle, with the adjacent side equaling 10, and the left angle of the adjacent angle equaling 45 degrees. the hypotenuse side is Y and the oppisite side is X. how do you find x and y, and what are they???
Well, since we have a 45-degree angle, it's party time – because we're dealing with a 45-45-90 triangle! In this special right triangle, the sides are always in a ratio of 1:1:√2.
So with the adjacent side equaling 10, we know that the opposite side (X) and the hypotenuse (Y) will be equal. Let's call them both "a."
Using the ratio, we have:
Adjacent side (10) : Opposite side (a) : Hypotenuse (a)
1 : 1 : √2
Now, let's solve for a:
1/1 = a/10 (cross-multiply) 1 × 10 = 1 × a 10 = a
So X = 10 and Y = 10 as well!
In conclusion, X (the opposite side) is equal to 10, and Y (the hypotenuse) is also equal to 10. It's like a pair of conjoined twins – they are inseparable!
To find the values of X and Y in the special right triangle, we can use trigonometric ratios. In this case, we are given the adjacent side and the angle.
1. Given: Adjacent side = 10, and angle = 45 degrees.
2. Since we know the adjacent side and the angle, we can use the trigonometric ratio tangent (tan) to find the opposite side (X).
Tangent of an angle = opposite side / adjacent side.
Tangent of 45 degrees = X / 10.
Here, the tangent of 45 degrees is 1 (as it is a special right triangle).
So, X / 10 = 1.
X = 10.
3. To find the hypotenuse side (Y), we can use the Pythagorean theorem.
According to the Pythagorean theorem, in a right triangle, the sum of the squares of the two legs (the two shorter sides) is equal to the square of the hypotenuse side (the longest side).
So, X^2 + Y^2 = 10^2 + Y^2 = Y^2 + 100 = Y^2.
Solving this equation, we find that Y = 10.
Therefore, X = 10 and Y = 10 in the given special right triangle.
To find the values of X and Y in the special right triangle, we can use the trigonometric ratios of sine, cosine, and tangent.
In this case, we have the given angle of 45 degrees and the adjacent side length of 10.
First, let's find the value of X (the opposite side):
We can use the sine ratio, which is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
sin(angle) = opposite/hypotenuse
In this case, sin(45 degrees) = X/Y
So, X/Y = sin(45 degrees)
To find the value of Y, we rearrange the equation:
Y = X / sin(45 degrees)
Now, we substitute the given value of X = 10:
Y = 10 / sin(45 degrees)
To find the value of Y, we can use a scientific calculator. By entering sin(45 degrees) into the calculator, we get the value of √2/2.
Y = 10 / (√2/2)
Y = 10 * (2/√2)
Y = 10 * (2√2/2)
Y = 10√2
So, the value of Y is 10√2.
Similarly, to find the value of X, we can use the cosine ratio, which is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
cos(angle) = adjacent/hypotenuse
In this case, cos(45 degrees) = 10/Y
So, 10/Y = cos(45 degrees)
Y = 10 / cos(45 degrees)
Again, we substitute the value of cos(45 degrees) into a scientific calculator to find its value, which is √2/2.
Y = 10 / (√2/2)
Y = 10 * (2/√2)
Y = 10 * (2√2/2)
Y = 10√2
So, the value of Y is also 10√2.
To summarize:
X = 10
Y = 10√2
Therefore, X is equal to 10, and Y is equal to 10√2 in the given special right triangle.