Question

Find sin2x, cos2x, and tan2x if tanx=-12/5 and x terminates in quadrant II

Answer 1

The values are:

`\[ \sin(2x) = -120 \]\\ \cos(2x) = -119 \]\\ \tan(2x) = (120)/(119) \]`

Given that
`\( \tan(x) = -(12)/(5) \)` and x terminates in quadrant II, you can use trigonometric identities to find
`\( \sin(2x) \), \( \cos(2x) \), and \( \tan(2x) \)`

1. Find
`\( \sin(x) \) and \( \cos(x) \):`

`Since \( \tan(x) = (\sin(x))/(\cos(x)) \), you can use the fact that \( \tan(x) = -(12)/(5) \) to find \( \sin(x) \) and \( \cos(x) \).`

`\[ \tan(x) = (\sin(x))/(\cos(x)) = -(12)/(5) \]`

From this, you can deduce that
`\( \sin(x) = -12 \) and \( \cos(x) = 5 \).`

2. Use double-angle formulas:

The double-angle formulas are:

`\[ \sin(2x) = 2\sin(x)\cos(x) \]\\ \cos(2x) = \cos^2(x) - \sin^2(x) \]\\ \tan(2x) = (\sin(2x))/(\cos(2x)) \]`

3. Calculate
`\( \sin(2x) \)`:

`\[ \sin(2x) = 2 \cdot (-12) \cdot 5 = -120 \]`

4. Calculate
`\( \cos(2x) \)`:

`\[ \cos(2x) = 5^2 - (-12)^2 = 25 - 144 = -119 \]`

5. Calculate
`\( \tan(2x) \)`:

`\[ \tan(2x) = (-120)/(-119) = (120)/(119) \]`

So, the values are:

`\[ \sin(2x) = -120 \]\\ \cos(2x) = -119 \]\\ \tan(2x) = (120)/(119) \]`