Question

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

Answer 1

`\sin 2x = (24)/(25)` ,
`\cos 2x = (7)/(25)`,
`\tan 2x = (24)/(7)`

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 = (2\cdot \tan x)/(1-\tan^(2)x)`

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

`\sin x = (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 = (y)/(r)`

`\tan x = (s)/(y)`

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

`\cos x = -(4)/(5)` and
`\tan x = (3)/(4)`.

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

`\sin 2x = 2\cdot \left(-(3)/(5) \right)\cdot \left(-(4)/(5) \right)`

`\sin 2x = (24)/(25)`

`\cos 2x = \left(-(4)/(5) \right)^(2)-\left(-(3)/(5) \right)^(2)`

`\cos 2x = (7)/(25)`

`\tan 2x = (2\cdot \left((3)/(4) \right))/(1-\left((3)/(4) \right)^(2))`

`\tan 2x = (24)/(7)`