Question

How do you find the perimeter of a triangle with 1 side given the angles are 105, 45 and 30 and the only side given is 8, opposite of the angle 30?

Answer 1

When we have two angles and one of the sides of a triangle, then we can find out all other information about that particular triangle by using the law of sines. Here, since we have all the three angles and measurement of one side, thus we shall apply the law of sines to find the measurements of the remaining two sides. Further, on adding all the three sides of the triangle, we shall calculate its perimeter.

Complete step-by-step answer:
The law of sines basically tells us that the ratio between the sine of an angle and the side opposite to that angle is going to be constant for any of the angles in a triangle.
According to law of sines,
$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$
Where, A, B, C are the three angles of the triangle and a, b, c are the three sides of the triangle opposite to the angles A, B, C respectively.
Here, we have $\angle A={{30}^{\circ }},\angle B={{45}^{\circ }},\angle C={{105}^{\circ }}$ and $a=8$ because it is the side opposite of the angle 30.
$\Rightarrow \dfrac{\sin {{30}^{\circ }}}{8}=\dfrac{\sin {{45}^{\circ }}}{b}=\dfrac{\sin {{105}^{\circ }}}{c}$
Now we shall take these in pairs and find sides b and c.
First let us take, $\dfrac{\sin {{30}^{\circ }}}{8}=\dfrac{\sin {{45}^{\circ }}}{b}$
$\Rightarrow \dfrac{\sin {{30}^{\circ }}}{8}=\dfrac{\sin {{45}^{\circ }}}{b}$
We know that $\sin {{30}^{\circ }}=0.5$ and $\sin {{45}^{\circ }}=0.707107$. Applying these values, we get
$\Rightarrow \dfrac{0.5}{8}=\dfrac{0.707107}{b}$
Cross-multiplying these values, we get
$\Rightarrow b=\dfrac{0.707107\times 8}{0.5}$
$\Rightarrow b=11.313712$
Now let us take, $\dfrac{\sin {{30}^{\circ }}}{8}=\dfrac{\sin {{105}^{\circ }}}{c}$
$\Rightarrow \dfrac{\sin {{30}^{\circ }}}{8}=\dfrac{\sin {{105}^{\circ }}}{c}$
We know that $\sin {{30}^{\circ }}=0.5$ and $\sin {{105}^{\circ }}=0.965926$. Applying these values, we get
$\Rightarrow \dfrac{0.5}{8}=\dfrac{0.965926}{c}$
Cross-multiplying these values, we get
$\Rightarrow c=\dfrac{0.965926\times 8}{0.5}$
$\Rightarrow c=15.454816$
Also, perimeter of triangle is equal to sum of all sides.
$\Rightarrow perimeter=a+b+c$
$\Rightarrow perimeter=8+11.313712+15.454816$
$\Rightarrow perimeter=34.768528$ units
Therefore, perimeter of a triangle with one side given the angles are 105, 45 and 30 and the only side given is 8, opposite of the angle 30 is 34.768528 units.

Note: The function used in the law of sines is a sine function which is a trigonometric function. The value of sine function for any angle is calculated by the Pythagoras theory. According to the Pythagorean theory, in a right-angled triangle, the sine of an angle is equal to the ratio of the perpendicular side (the side opposite to the right angle) and the hypotenuse.