in Triangle abc, tanA=3/4, tan B=1, and a=10. find b in simplest radical form.
3,4,5 triangle
sin A = 3/5
1,1,sqrt 2 triangle
sin B = sqrt 2/2
law of sines
sin A/10 = sin B/b
To find the length of side b in Triangle ABC, we can use the tangent function and the given information. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
Given that tan(A) = 3/4, we can determine that the opposite side to angle A is 3x and the adjacent side is 4x (where x is a positive constant).
Knowing that the length of side a is 10, we can set up the following equation based on the Pythagorean theorem:
(3x)^2 + (4x)^2 = 10^2
9x^2 + 16x^2 = 100
25x^2 = 100
Divide both sides of the equation by 25:
x^2 = 4
Taking the square root of both sides, we have:
x = 2
Now, we can find the length of side b by multiplying x by the adjacent side ratio:
b = 4x = 4(2) = 8
Therefore, the length of side b is 8.
To find the length of side b in triangle ABC, you can use the tangent ratio:
tan(A) = opposite/adjacent
Given that tan(A) = 3/4, we know that the opposite side length is 3 and the adjacent side length is 4.
Similarly, tan(B) = 1, which means the opposite side length is 1 and the adjacent side length is 1.
Now, we can use the Law of Sines to establish a relationship between the sides and angles in a triangle:
a/sin(A) = b/sin(B)
Plugging in the known values, we have:
10/sin(A) = b/sin(B)
We can rewrite sin(A) and sin(B) in terms of their corresponding ratios:
10/(3/5) = b/(1/1)
Simplifying, we get:
10 * (5/3) = b * (1/1)
50/3 = b
Therefore, the length of side b is 50/3 in simplest radical form.