In triangular ABC, AB =6cm BC=7cm and AC=8cm. What is the difference between the largest and the smallest angle of a triangle.
You know that angle B is the greatest (since b is the longest side)
and angle C is the smallest, since c is the shortest side.
find A using law of cosines:
b^2 = a^2 + c^2 - 2ac cosB
Find C the same way, or using the law of sines:
sinC/c = sinB/b
Now just find B-C
To find the difference between the largest and smallest angle of a triangle, we need to determine the measures of all three angles.
In a triangle, the sum of the three angles is always 180 degrees.
Let's label the angles as A, B, and C for convenience, corresponding to vertices A, B, and C of triangle ABC.
Given that sides AB, BC, and AC are 6cm, 7cm, and 8cm, respectively, we can use the Law of Cosines to find the measures of angles A, B, and C.
The Law of Cosines states that for any triangle with sides a, b, and c, and opposite angles A, B, and C, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
Now let's plug in the values for triangle ABC:
Angle A:
a = BC = 7cm
b = AC = 8cm
c = AB = 6cm
Using the Law of Cosines, we can find cos(A):
6^2 = 8^2 + 7^2 - 2 * 8 * 7 * cos(A)
36 = 64 + 49 - 112 * cos(A)
-77 = -112 * cos(A)
cos(A) = -77 / -112
cos(A) = 0.6875
Taking the inverse cosine (cos^(-1)) of 0.6875, we can find the measure of angle A:
A = cos^(-1)(0.6875)
A ≈ 46.6 degrees
Similarly, we can find the measures of angles B and C:
Angle B:
a = AC = 8cm
b = AB = 6cm
c = BC = 7cm
Using the Law of Cosines and following the same steps as above, we find:
B ≈ 64.5 degrees
Angle C:
a = AB = 6cm
b = BC = 7cm
c = AC = 8cm
Using the Law of Cosines and following the same steps as above, we find:
C ≈ 69 degrees
Now we have the measures of angles A, B, and C. We can determine the difference between the largest and smallest angles by subtracting the smallest angle (angle A) from the largest angle (angle C):
Difference = C - A
Difference ≈ 69 - 46.6
Difference ≈ 22.4 degrees
Therefore, the difference between the largest and smallest angles of triangular ABC is approximately 22.4 degrees.