Answer:
,
, ![\tan 2x = \frac{24}{7}](https://tex.z-dn.net/?f=%5Ctan%202x%20%3D%20%5Cfrac%7B24%7D%7B7%7D)
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](https://tex.z-dn.net/?f=%5Csin%202x%20%3D%202%5Ccdot%20%5Csin%20x%20%5Ccdot%20%5Ccos%20x)
![\cos 2x = \cos^{2}x -\sin^{2}x](https://tex.z-dn.net/?f=%5Ccos%202x%20%3D%20%5Ccos%5E%7B2%7Dx%20-%5Csin%5E%7B2%7Dx)
![\tan 2x = \frac{2\cdot \tan x}{1-\tan^{2}x}](https://tex.z-dn.net/?f=%5Ctan%202x%20%3D%20%5Cfrac%7B2%5Ccdot%20%5Ctan%20x%7D%7B1-%5Ctan%5E%7B2%7Dx%7D)
According to the definition of sine function, the ratio is represented by:
![\sin x = \frac{s}{r}](https://tex.z-dn.net/?f=%5Csin%20x%20%3D%20%5Cfrac%7Bs%7D%7Br%7D)
Where:
- Opposite leg, dimensionless.
- Hypotenuse, dimensionless.
Since
, measured in sexagesimal degrees, is in third quadrant, the following relation is known:
and
.
Where
is represented by the Pythagorean identity:
The magnitude of
is found by means the Pythagorean expression:
![r^{2} = s^{2}+y^{2}](https://tex.z-dn.net/?f=r%5E%7B2%7D%20%3D%20s%5E%7B2%7D%2By%5E%7B2%7D)
![y^{2} = r^{2}-s^{2}](https://tex.z-dn.net/?f=y%5E%7B2%7D%20%3D%20r%5E%7B2%7D-s%5E%7B2%7D)
![y = \sqrt{r^{2}-s^{2}}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%7Br%5E%7B2%7D-s%5E%7B2%7D%7D)
Where
is the adjacent leg, dimensionless.
If
and
, the value of
is:
Then, the definitions for cosine and tangent of x are, respectively:
![\cos x = \frac{y}{r}](https://tex.z-dn.net/?f=%5Ccos%20x%20%3D%20%5Cfrac%7By%7D%7Br%7D)
If
,
and
, the values for each identity are, respectively:
and
.
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)](https://tex.z-dn.net/?f=%5Csin%202x%20%3D%202%5Ccdot%20%5Cleft%28-%5Cfrac%7B3%7D%7B5%7D%20%5Cright%29%5Ccdot%20%5Cleft%28-%5Cfrac%7B4%7D%7B5%7D%20%5Cright%29)
![\cos 2x = \left(-\frac{4}{5} \right)^{2}-\left(-\frac{3}{5} \right)^{2}](https://tex.z-dn.net/?f=%5Ccos%202x%20%3D%20%5Cleft%28-%5Cfrac%7B4%7D%7B5%7D%20%5Cright%29%5E%7B2%7D-%5Cleft%28-%5Cfrac%7B3%7D%7B5%7D%20%5Cright%29%5E%7B2%7D)
![\cos 2x = \frac{7}{25}](https://tex.z-dn.net/?f=%5Ccos%202x%20%3D%20%5Cfrac%7B7%7D%7B25%7D)
![\tan 2x = \frac{2\cdot \left(\frac{3}{4} \right)}{1-\left(\frac{3}{4} \right)^{2}}](https://tex.z-dn.net/?f=%5Ctan%202x%20%3D%20%5Cfrac%7B2%5Ccdot%20%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20%5Cright%29%7D%7B1-%5Cleft%28%5Cfrac%7B3%7D%7B4%7D%20%5Cright%29%5E%7B2%7D%7D)
![\tan 2x = \frac{24}{7}](https://tex.z-dn.net/?f=%5Ctan%202x%20%3D%20%5Cfrac%7B24%7D%7B7%7D)