Question

Suppose tan(x) = 2/3, and the terminal side of x is located in quadrant I. What is sin(x)?

A. 2/sqr13
B. 3/sqr13
C. 3/2
D. sqr13 /2

Answer 1

Using trigonometric relations, it is found that option A is correct, that is:

$\sin{x} = \frac{2}{\sqrt{13}}$

---------------------

Tangent of an angle is sine of this angle divided by the cosine of this angle, thus:

$\tan{x} = \frac{\sin{x}}{\cos{x}}$

Another important relation is:

$\sin^2{x} + \cos^2{x} = 1$

Tangent is 2/3, that is:

$\tan{x} = \frac{2}{3}$

Which means that:

$\frac{\sin{x}}{\cos{x}} = \frac{2}{3}$

We want to find the sine, so we write the cosine as a function of the sine, that is:

$2\cos{x} = 3\sin{x}$

$\cos{x} = \frac{3\sin{x}}{2}$

Replacing in the equation for the relation:

$\sin^2{x} + \cos^2{x} = 1$

$\sin^2{x} + (\frac{3\sin{x}}{2})^2 = 1$

$\sin^2{x} + \frac{9\sin^2{x}}{4} = 1$

$\frac{13\sin^{2}{x}}{4} = 1$

$\sin^2{x} = \frac{4}{13}$

$\sin{x} = \pm \sqrt{\frac{4}{13}}$

First quadrant, thus positive:

$\sin{x} = \frac{2}{\sqrt{13}}$

A similar problem is given at https://brainly.com/question/24028133