If $\tan \theta = \frac{3}{4}$, find the value of $\sin \theta + \cos \theta$.

1. Analyze the question

We are given the value of $\tan \theta$ as a fraction and asked to find the sum of $\sin \theta$ and $\cos \theta$. To solve this, we need to determine the lengths of all three sides of a right-angled triangle using the given trigonometric ratio.

2. Explain the concept

In a right-angled triangle, trigonometric ratios are defined using the mnemonic SOH CAH TOA:

• 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 side = 3
• Adjacent side = 4

We use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the Hypotenuse ($h$): $h^2 = 3^2 + 4^2$

3. Show the solution

Step 1: Find the Hypotenuse $h^2 = 9 + 16$ $h^2 = 25$ $h = \sqrt{25} = 5$

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 the sum $\sin \theta + \cos \theta = \frac{3}{5} + \frac{4}{5}$ $\sin \theta + \cos \theta = \frac{7}{5}$

Step 4: Convert to a mixed fraction $\frac{7}{5} = 1 \frac{2}{5}$

4. Identify the answer

Comparing our result to the given options: A. $1\frac{1}{3}$ B. $1\frac{2}{3}$ C. $1\frac{3}{5}$ D. $1\frac{2}{5}$

The calculated value matches option D.

5. Tip

Whenever you see $\tan \theta = \frac{3}{4}$, immediately recognize the 3-4-5 Pythagorean triple. This is a common triangle in JAMB questions; knowing these triples (like 5-12-13 or 8-15-17) saves you time during the exam because you won't have to calculate the hypotenuse from scratch!