Two sides of a triangle are 50 m and 60 m long. the angle included between these sides is 30 deg. what is the interior angle opposite the longest side?
Answers
Given:
The lengths of the two sides of a triangle are 50 m and 60 m
The value of the included angle is 30°.
To find:
The value of interior angle opposite to the longest side
Solution:
To solve this we will use the cosine rule or the law of cosines,
which states that the square of the third side of a triangle is the sum of the squares of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.
Assuming,
A= 50 cm
B = 60 cm
∠C = 30°
c²=a²+b²-2abcosC.
Substituting the values
c²=(50)²+(60)²-2(50)(60)cos(30)
c = 30.06 m
According to this, B which is equal to 60 cm is the longest side
Now let's find the angle opposite to this side, ∠C
Using the law of sines that states that the ratio of the side and the corresponding angle of a triangle is equal to the ratio of other sides and their corresponding angles.
A/sin∠A = B/sin∠B = C/sin∠C
B/sin∠B = C/sin∠C
substituting values
60/ sin∠B = 30.06/sin (30°)
sin∠B = 60 × sin (30°) / 30.06
sin∠B = 60 × 0.5 / 30.06
sin∠B = 30/ 30.06
If we approximate the
sin∠B = 1
∠B = 90°
But we if take precise value then the answer
sin∠B = 0.998
∠B = 86.38°
Therefore, the interior angle opposite the longest side is 86.38°.