Given tan A = 4 3 , find the other trigonometric ratios of the angle A.
Step-by-Step Solution
Key Concept: Use the definition of tangent in a right‑angled triangle (tan A = opposite/adjacent). Choose convenient integer lengths for the opposite and adjacent sides, apply the Pythagorean theorem to obtain the hypotenuse, and then use the reciprocal relations to obtain the remaining trigonometric ratios.
1. From the definition, \(\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}\).
Choose a right‑angled triangle with opposite side = 4 units and adjacent side = 3 units.\
2. Compute the hypotenuse using Pythagoras: \[\text{hypotenuse} = \sqrt{4^{2}+3^{2}} = \sqrt{16+9}=\sqrt{25}=5\]
3. Find \(\sin A\) and \(\cos A\):\
\[\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}, \qquad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}\]
4. Obtain the remaining ratios using reciprocal identities:\
\[\cot A = \frac{1}{\tan A}=\frac{3}{4},\qquad \sec A = \frac{1}{\cos A}=\frac{5}{3},\qquad \csc A = \frac{1}{\sin A}=\frac{5}{4}\]
5. Summarise all the trigonometric ratios of \(A\):\
\[\tan A = \frac{4}{3},\; \cot A = \frac{3}{4},\; \sin A = \frac{4}{5},\; \cos A = \frac{3}{5},\; \sec A = \frac{5}{3},\; \csc A = \frac{5}{4}\]
Correct Answer: sin A = 4/5, cos A = 3/5, tan A = 4/3 (given), cot A = 3/4, sec A = 5/3, cosec A = 5/4