Question

In the adjoining figure, $ \text{AD}=4\text{ cm} $ , $ \text{BD}=3\text{ cm} $ and $ \text{CB}=12\text{ cm} $ . Find $ \cot \theta $ .

Answer 1

Hint:
 We have two right-angled triangles. In one triangle, we know the measurements of two sides. We will use the Pythagoras theorem to find the length of the third side. After that, we will have the lengths of two sides of the other right angled triangle. We will use the definition of the trigonometric function $ \cot \theta $ to obtain the desired value.

Complete step by step answer:
Let us consider $ \Delta \text{ABD} $ . We have $ \text{AD}=4\text{ cm} $ , $ \text{BD}=3\text{ cm} $ and $ \angle \text{ADB}=90{}^\circ $ . We will use the Pythagoras theorem to find the length of the hypotenuse AB. The Pythagoras theorem states that
 $ {{\left( \text{hypotenuse} \right)}^{2}}={{\left( \text{side 1} \right)}^{2}}+{{\left( \text{side 2} \right)}^{2}} $
Substituting the sides of $ \Delta \text{ABD} $ , we get
 $ {{\left( \text{AB} \right)}^{2}}={{\left( \text{AD} \right)}^{2}}+{{\left( \text{BD} \right)}^{2}} $
Therefore, substituting the lengths, we have the following
 $ \begin{align}
  & {{\left( \text{AB} \right)}^{2}}={{\left( 4 \right)}^{2}}+{{\left( \text{3} \right)}^{2}} \\
 & \Rightarrow {{\left( \text{AB} \right)}^{2}}=16+9 \\
 & \Rightarrow {{\left( \text{AB} \right)}^{2}}=25 \\
 & \therefore \text{AB}=5\text{ cm} \\
\end{align} $
Now, let us consider $ \Delta \text{ABC} $ . We have $ \text{AB}=5\text{ cm} $ , $ \text{CB}=12\text{ cm} $ , $ \angle \text{ABC}=90{}^\circ $ and $ \angle \text{ACB}=\theta $ . In a right angled triangle the definition of the trigonometric function $ \cot \theta $ is the following,
 $ \cot \theta =\dfrac{\text{Adjacent}}{\text{Opposite}} $
The adjacent side to the angle $ \theta $ in $ \Delta \text{ABC} $ is CB and the opposite side is AB. Hence, we have
 $ \cot \theta =\dfrac{\text{CB}}{\text{AB}} $
Substituting the values of the lengths of the sides, we get
 $ \cot \theta =\dfrac{12}{5} $ .

Note:
We should be aware of the properties of a right-angled triangle for this type of question. The figure and the given information are the biggest hints for us to choose the correct property or formulae that can be used in order to get the required answer. The knowledge of trigonometric functions and their definitions is necessary to obtain the desired answer. We can derive the definition of all trigonometric functions even if we know only the definitions of the sine and cosine functions. But for this, we should understand the relations among the trigonometric functions.