Question

If $\tan \theta = \tfrac{3}{4}$, then the value of $\cos \theta$ will be:
 

• A. $\tfrac{4}{5}$
• B. $\tfrac{3}{5}$
• C. $\tfrac{4}{3}$
• D. $\tfrac{5}{4}$

Hint:

Whenever $\tan \theta$ is given, you can imagine a right-angled triangle with Perpendicular = numerator and Base = denominator. Then apply Pythagoras theorem to find the hypotenuse, which makes it easy to calculate $\sin \theta$ or $\cos \theta$.

Answer 1

The Correct Option is A

Step 1: Recall the definition of tangent
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} \] Here, $\tan \theta = \tfrac{3}{4}$ means: - Perpendicular (opposite side) = 3 units
- Base (adjacent side) = 4 units

Step 2: Use Pythagoras theorem to find the hypotenuse
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \] \[ \text{Hypotenuse}^2 = 3^2 + 4^2 = 9 + 16 = 25 \] \[ \text{Hypotenuse} = \sqrt{25} = 5 \]

Step 3: Write the formula for cosine
\[ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{4}{5} \]

Step 4: Verify against options
The value of $\cos \theta$ is $\tfrac{4}{5}$, which matches option (A).
\[ \boxed{\cos \theta = \tfrac{4}{5}} \]