An angle opposite of a leg of a right triangle measures w, and tan(w)= 3/4.
What is sin(90-w)?
hypotenuse = 5 (3,4,5 right triangle)
sin (90-w) = cos w = 4/5
Well, since we're dealing with a right triangle and one angle is opposite a leg, we can use the tangent function to find the value of w. If tan(w) = 3/4, that means the ratio of the length of the opposite side to the adjacent side is 3/4.
Now, let's find the length of the hypotenuse using the Pythagorean Theorem. Since the triangle is a right triangle, we have:
(adjacent side)^2 + (opposite side)^2 = (hypotenuse)^2
Let's say the adjacent side is x, then we have:
x^2 + (3/4)^2x^2 = (hypotenuse)^2
Simplifying the equation:
1^2x^2 + (3/4)^2x^2 = (hypotenuse)^2
16/16x^2 + 9/16x^2 = (hypotenuse)^2
25/16x^2 = (hypotenuse)^2
Since (hypotenuse)^2 = 25/16x^2, we can take the square root of both sides:
hypotenuse = √(25/16x^2)
Therefore, the hypotenuse is (5/4)x.
Now, we want to find sin(90-w), which is the ratio of the length of the opposite side to the hypotenuse. So, sin(90-w) = (opposite side)/(hypotenuse).
Using the given information, the opposite side of the triangle is w, and the hypotenuse is (5/4)x.
Hence, sin(90-w) = w/((5/4)x).
To find sin(90-w), we first need to find the value of w.
Given that tan(w) = 3/4, we can find w by taking the arctan of (3/4) :
w = arctan(3/4)
Next, we can find sin(90-w) using the following identity:
sin(90-w) = cos(w)
Therefore, we need to find cos(w). To do this, we can use the Pythagorean theorem, as the triangle in question is a right triangle:
cos(w) = adjacent/hypotenuse
Since we know that tan(w) = 3/4, we can label the adjacent side as 3 and the opposite side as 4. By using the Pythagorean theorem, we can find the hypotenuse:
hypotenuse = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
Now we can find cos(w):
cos(w) = 3/5
Therefore, sin(90-w) = cos(w) = 3/5.
To find sin(90-w), we need to first find the value of w.
Given that tan(w) = 3/4, we can use the inverse tangent function (arctan) to find the angle w. The arctan function will give us the angle in radians, so we will need to convert it to degrees later.
arctan(3/4) ≈ 36.87 degrees
Now that we know the value of w, we can subtract it from 90 degrees to find 90-w.
90 - 36.87 ≈ 53.13 degrees
Now we can find sin(90-w) by taking the sine of 53.13 degrees.
sin(53.13) ≈ 0.799
Therefore, sin(90-w) is approximately 0.799.