Answer:
[tex]\sin 2x = \frac{24}{25}[/tex] , [tex]\cos 2x = \frac{7}{25}[/tex], [tex]\tan 2x = \frac{24}{7}[/tex]
Step-by-step explanation:
The sine, cosine and tangent of a double angle are given by the following trigonometric identities:
[tex]\sin 2x = 2\cdot \sin x \cdot \cos x[/tex]
[tex]\cos 2x = \cos^{2}x -\sin^{2}x[/tex]
[tex]\tan 2x = \frac{2\cdot \tan x}{1-\tan^{2}x}[/tex]
According to the definition of sine function, the ratio is represented by:
[tex]\sin x = \frac{s}{r}[/tex]
Where:
[tex]s[/tex] - Opposite leg, dimensionless.
[tex]r[/tex] - Hypotenuse, dimensionless.
Since [tex]x[/tex], measured in sexagesimal degrees, is in third quadrant, the following relation is known:
[tex]s < 0[/tex] and [tex]y < 0[/tex].
Where [tex]r[/tex] is represented by the Pythagorean identity:
[tex]r = \sqrt{s^{2}+y^{2}}[/tex]
The magnitude of [tex]y[/tex] is found by means the Pythagorean expression:
[tex]r^{2} = s^{2}+y^{2}[/tex]
[tex]y^{2} = r^{2}-s^{2}[/tex]
[tex]y = \sqrt{r^{2}-s^{2}}[/tex]
Where [tex]y[/tex] is the adjacent leg, dimensionless.
If [tex]s = -3[/tex] and [tex]r = 5[/tex], the value of [tex]y[/tex] is:
[tex]y = \sqrt{(5^{2})-(-3)^{2}}[/tex]
[tex]y = -4[/tex]
Then, the definitions for cosine and tangent of x are, respectively:
[tex]\cos x = \frac{y}{r}[/tex]
[tex]\tan x = \frac{s}{y}[/tex]
If [tex]s = -3[/tex], [tex]y = -4[/tex] and [tex]r = 5[/tex], the values for each identity are, respectively:
[tex]\cos x = -\frac{4}{5}[/tex] and [tex]\tan x = \frac{3}{4}[/tex].
Now, the value for each double angle identity are obtained below:
[tex]\sin 2x = 2\cdot \left(-\frac{3}{5} \right)\cdot \left(-\frac{4}{5} \right)[/tex]
[tex]\sin 2x = \frac{24}{25}[/tex]
[tex]\cos 2x = \left(-\frac{4}{5} \right)^{2}-\left(-\frac{3}{5} \right)^{2}[/tex]
[tex]\cos 2x = \frac{7}{25}[/tex]
[tex]\tan 2x = \frac{2\cdot \left(\frac{3}{4} \right)}{1-\left(\frac{3}{4} \right)^{2}}[/tex]
[tex]\tan 2x = \frac{24}{7}[/tex]
