The three angles of a triangle are 105°,25°,50° .
In what ratios are the sides?
21:5:10
How do you know that?
I divided each by the common factor.
In this case. Would you use sinA/a=sinB/b=sinC/c
To determine the ratios of the sides of a triangle with given angles, we can use the Law of Sines. According to the Law of Sines, the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of a triangle.
Let's label the three angles of the triangle as A, B, and C, with their corresponding sides as a, b, and c. Therefore, we have:
- Angle A = 105°, side a is opposite to angle A.
- Angle B = 25°, side b is opposite to angle B.
- Angle C = 50°, side c is opposite to angle C.
Now, let's apply the Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
Substituting the given values:
a/sin(105°) = b/sin(25°) = c/sin(50°)
To find the ratios of the sides, we can use the values of the trigonometric functions sin(105°), sin(25°), and sin(50°). Using a scientific calculator or an online calculator, evaluate the sine of each angle:
sin(105°) ≈ 0.9659258262890683
sin(25°) ≈ 0.42261826174069944
sin(50°) ≈ 0.766044443118978
Now, substitute these values back into the Law of Sines equation:
a/0.9659258262890683 = b/0.42261826174069944 = c/0.766044443118978
Therefore, the sides are in the following ratios:
a : b : c ≈ 0.9659258262890683 : 0.42261826174069944 : 0.766044443118978
Simplifying, we get:
a : b : c ≈ 2.288245611 : 1 : 1.455053
Hence, the sides of the triangle are in the ratios of approximately 2.288245611 : 1 : 1.455053.