In a right angled triangle, if tan $\theta$ = $\frac{3}{4}$. What is cos$\theta$ - sin$\theta$?

1. Analyze the question

We are given the value of $\tan \theta$ in a right-angled triangle and asked to find the value of the expression $(\cos \theta - \sin \theta)$.

2. Explain the concept

In a right-angled triangle, we use the SOH CAH TOA mnemonic:

• SOH: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
• CAH: $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
• TOA: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

Given $\tan \theta = \frac{3}{4}$, we identify:

• Opposite = 3
• Adjacent = 4

To find the Hypotenuse, we use the Pythagoras theorem: $\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2$

3. Show the solution

Step 1: Find the Hypotenuse. $H^2 = 3^2 + 4^2$ $H^2 = 9 + 16$ $H^2 = 25$ $H = \sqrt{25} = 5$

Step 2: Determine $\sin \theta$ and $\cos \theta$.

• $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}$
• $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}$

Step 3: Calculate $\cos \theta - \sin \theta$. $\cos \theta - \sin \theta = \frac{4}{5} - \frac{3}{5}$ $\cos \theta - \sin \theta = \frac{4 - 3}{5} = \frac{1}{5}$

4. Identify the answer

Comparing our result to the given options, the correct choice is C.

5. Tip

Whenever you see a trigonometric ratio involving 3 and 4, remember the 3-4-5 Pythagorean triple. It is a common pattern in JAMB questions that saves you time from calculating the hypotenuse from scratch!