Understand the Question We are given a right-angled triangle , right-angled at , with and hypotenuse . First, find the length of the unknown side using the Pythagoras theorem: Next, find and using trigonometric ratios: For angle : For angle : Finally, subtract from . Step 1 · Find Side QR In right-angled triangle , right-angled at : By Pythagoras theorem Step 2 · Calculate and For angle : For angle : Step 3 · Evaluate Substitute the values of and Answer > Common Mistakes Angle Reference Confusion: The opposite and adjacent sides change depending on the angle: For , the opposite side is and the adjacent side is . For , the opposite side is and the adjacent side is . Pythagoras Subtraction Error: Adding instead of subtracting when finding a leg: , not .

Introduction to Trigonometry | Exercise 8.1

Question 2

In Fig. 8.13, find $\tan P - \cot R$ .

Solution

Understand the Question

We are given a right-angled triangle $\text{PQR}$ , right-angled at $\text{Q}$ , with $\text{PQ} = 12\text{ cm}$ and hypotenuse $\text{PR} = 13\text{ cm}$ .
First, find the length of the unknown side $\text{QR}$ using the Pythagoras theorem: $\text{PR}^2 = \text{PQ}^2 + \text{QR}^2$
Next, find $\tan P$ $tan P$ and $\cot R$ $cot R$ using trigonometric ratios:
- For angle $\text{P}$ : $\tan P = \dfrac{\text{Opposite}}{\text{Adjacent}} = \dfrac{\text{QR}}{\text{PQ}}$
- For angle $\text{R}$ : $\cot R = \dfrac{\text{Adjacent}}{\text{Opposite}} = \dfrac{\text{QR}}{\text{PQ}}$
Finally, subtract $\cot R$ from $\tan P$ .

Step 1 · Find Side QR

In right-angled triangle $\text{PQR}$ , right-angled at $\text{Q}$ : Diagram 1

By Pythagoras theorem

\begin{aligned} \text{PR}^2 &= \text{PQ}^2 + \text{QR}^2 \\ 13^2 &= 12^2 + \text{QR}^2 \\ 169 &= 144 + \text{QR}^2 \\ \text{QR}^2 &= 169 - 144 \\ \text{QR}^2 &= 25 \\ \text{QR} &= \sqrt{25} = 5\text{ cm} \end{aligned}

Step 2 · Calculate $\tan P$ and $\cot R$

For angle $\text{P}$ : $\tan P = \dfrac{\text{Opposite side}}{\text{Adjacent side}} = \dfrac{\text{QR}}{\text{PQ}} = \dfrac{5}{12}$

For angle $\text{R}$ : $\cot R = \dfrac{\text{Adjacent side}}{\text{Opposite side}} = \dfrac{\text{QR}}{\text{PQ}} = \dfrac{5}{12}$

Step 3 · Evaluate $\tan P - \cot R$

Substitute the values of $\tan P$ and $\cot R$

\begin{aligned} \tan P - \cot R &= \dfrac{5}{12} - \dfrac{5}{12} \\[0.6em] &= 0 \end{aligned}

Answer

$0$

Common Mistakes

Angle Reference Confusion: The opposite and adjacent sides change depending on the angle:
- For $\angle P$ , the opposite side is $\text{QR}$ and the adjacent side is $\text{PQ}$ .
- For $\angle R$ , the opposite side is $\text{PQ}$ and the adjacent side is $\text{QR}$ .
Pythagoras Subtraction Error: Adding instead of subtracting when finding a leg: $\text{QR}^2 = \text{PR}^2 - \text{PQ}^2$ , not $\text{PR}^2 + \text{PQ}^2$ .

Question 2

More questions in Exercise 8.1