Question

If 3 tan A=4, then find sin A and cos A.

Answer 1

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Answer 2

Given:

3 tan A = 4

To find:

sin A =?

cos A =?

Solution:

To find the solution we need to write the ratio in terms of trigonometry for a right-angled triangle to get the measure of all the sides.

Step 1: Write trignometric ratio

tan θ = $\frac{perpendicular}{base}$

sin θ = $\frac{perpendicular}{hypotenuse}$

cos θ = $\frac{base}{hypotenuse}$

tan A = $\frac{perpendicular}{base}$=  $\frac{4}{3}$ ( ∵θ = ∠A)

Step 2: Calculate the measure of the unknown side

Hypotenuse can be calculated from the Pythagoras theorem for right-angled triangles.

hypotenuse of triangle = $\sqrt {opposite side^{2} + adjacent side^{2}$

hypotenuse  of triangle= $\sqrt{4^{2} + 3^{2} }$

hypotenuse of triangle = $\sqrt{16 + 9}$

hypotenuse of triangle = $\sqrt{25}$

hypotenuse of triangle  = 5

Step 3: Find value of other trignometric ratio

tan A = $\frac{opposite side }{adjacent side}$ = $\frac{4}{3}$ (given)

∴ sin A = $\frac{opposite side of traingle}{hypotenuse}$ = $\frac{4}{5}$

∴ cos A = $\frac{adjacent side of triangle}{hypotenuse}$ = $\frac{3}{5}$

Hence, value for sin A = $\frac{4}{5}$ and cos A = $\frac{3}{5}$.