Question

Find sin 2x, cos 2x, and tan 2x from the given information. sin x = -3/5, x in quadrant 3.

Answer 1

Answer:

$\sin 2x = \frac{24}{25}$ , $\cos 2x = \frac{7}{25}$, $\tan 2x = \frac{24}{7}$

Step-by-step explanation:

The sine, cosine and tangent of a double angle are given by the following trigonometric identities:

$\sin 2x = 2\cdot \sin x \cdot \cos x$

$\cos 2x = \cos^{2}x -\sin^{2}x$

$\tan 2x = \frac{2\cdot \tan x}{1-\tan^{2}x}$

According to the definition of sine function, the ratio is represented by:

$\sin x = \frac{s}{r}$

Where:

$s$ - Opposite leg, dimensionless.

$r$ - Hypotenuse, dimensionless.

Since $x$, measured in sexagesimal degrees, is in third quadrant, the following relation is known:

$s < 0$ and $y < 0$.

Where $r$ is represented by the Pythagorean identity:

$r = \sqrt{s^{2}+y^{2}}$

The magnitude of $y$ is found by means the Pythagorean expression:

$r^{2} = s^{2}+y^{2}$

$y^{2} = r^{2}-s^{2}$

$y = \sqrt{r^{2}-s^{2}}$

Where $y$ is the adjacent leg, dimensionless.

If $s = -3$ and $r = 5$, the value of $y$ is:

$y = \sqrt{(5^{2})-(-3)^{2}}$

$y = -4$

Then, the definitions for cosine and tangent of x are, respectively:

$\cos x = \frac{y}{r}$

$\tan x = \frac{s}{y}$

If $s = -3$, $y = -4$ and $r = 5$, the values for each identity are, respectively:

$\cos x = -\frac{4}{5}$ and $\tan x = \frac{3}{4}$.

Now, the value for each double angle identity are obtained below:

$\sin 2x = 2\cdot \left(-\frac{3}{5} \right)\cdot \left(-\frac{4}{5} \right)$

$\sin 2x = \frac{24}{25}$

$\cos 2x = \left(-\frac{4}{5} \right)^{2}-\left(-\frac{3}{5} \right)^{2}$

$\cos 2x = \frac{7}{25}$

$\tan 2x = \frac{2\cdot \left(\frac{3}{4} \right)}{1-\left(\frac{3}{4} \right)^{2}}$

$\tan 2x = \frac{24}{7}$