Question

How do you find the angle of a right-angled triangle when the hypotenuse and adjacent sides are given $HYP = 14cm$ , $ADJ = 6cm$ ?

Answer 1

Hint: Here, the given triangle is a right-angled triangle, which means that we can use the sine, cosine, and tangent functions to find the angle. As we are given the hypotenuse and the adjacent side to the angle here, we should use the cosine function to find the angle.

Complete step-by-step answer:
Here, a diagram is shown to represent the question.
In a right-angled triangle, a hypotenuse is the side opposite to the right angle and it is the longest side of the triangle.
In the triangle, one of the angles is equal to ${90^\circ }$.
For the other two angles, out of the two line segments that make the angle, one of them is the hypotenuse and the other line segment is called the adjacent side. And the remaining side is known as the opposite side as it is opposite to the angle.
Also as per the quadrilateral rule, we know that the total of all angles in a triangle is equal to ${180^\circ }$ as one angle is equal to ${90^\circ }$, the other two angles have to be less than ${90^\circ }$ to form a triangle.
Now, let’s suppose the angle we need to find as $\theta $.
Here, the length of the sides are given as
Hypotenuse = $\;14cm$
Adjacent side = $\;6cm$
From the trigonometric functions of sine, cosine, and tangent
To find $\sin \theta $, we require the opposite side and hypotenuse.
To find $\cos \theta $, we require the adjacent side and hypotenuse.
To find $\tan \theta $, we require the adjacent side and opposite side.
Here, the adjacent side and hypotenuse are given:
Hence, it is obvious that using $\cos \theta $ would be the best method.
Now, the formula for $\cos \theta $ is as follows,
$ \Rightarrow \cos \theta = \dfrac{{{\text{Adjacent side}}}}{{{\text{Hypotenuse}}}}$
Now, using the given values,
$ \Rightarrow \cos \theta = \dfrac{6}{{14}}$
The above fraction can be written in decimal as
$ \Rightarrow \cos \theta = 0.43$
Now using the inverse trigonometric equations we can write

$ \Rightarrow \theta = {\cos ^{ - 1}}(0.43)$
This is the value of the angle we need.

Additional information: With the use of a scientific calculator, the value of the angle found is ${64.53^\circ }$.

Note:
We can also find the angle by using the sine or tangent function. For that, first, we need to use the Pythagoras theorem $A{B^2} + B{C^2} = A{C^2}$. With the Pythagoras theorem, the length of the opposite side is found to be $4\sqrt {10} $. With this value, we can use both the sine or tangent function to find the angle $\theta $.