in triangle ABC, a = 2, b = the square root of 3, and m,angle C = 30 degrees. What is the value of c?
a= 1 b= 2 c= 2square root of 3 d=square root of 3 over 3
For your first one,
You should recognize those numbers as belonging to the 30-60-90° right-angled triangle
which would make c = 1
If you didn't notice that, you could use the cosine law
c^2 = 2^2 + √3^2 - 2(2)(√)cos30°
= 4 + 3 - 4(√3)(√3/2)
= 7 - 6
c^2 = 1
c = 1
Now, of course , in a triangle with sides, 1 , √3, and 2
2^2 = 1^2 + √3^2
so the triangle is right-angled, etc
To find the value of side length c in triangle ABC, we can use the law of cosines.
The law of cosines states that in a triangle with side lengths a, b, and c, and opposite angles A, B, and C respectively, the following equation holds true:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we are given the values a = 2, b = √3, and C = 30 degrees.
Substituting these values into the equation, we get:
c^2 = 2^2 + (√3)^2 - 2*2*√3*cos(30)
Simplifying this equation, we have:
c^2 = 4 + 3 - 4√3 * (1/2)
c^2 = 7 - 2√3 (since cos(30) = 1/2)
To find the exact value of c, we need to simplify further:
Take the square root of both sides:
√(c^2) = √(7 - 2√3)
c = ± √(7 - 2√3)
Therefore, c can have two possible values:
c = √(7 - 2√3) or c = -√(7 - 2√3)
Please note that the negative value is not considered in this context, as side lengths cannot be negative in a triangle.