Sure, let's tackle this problem step by step.
### Part a) Drawing a sketch that represents angle [tex]\( x \)[/tex]:
1. Identify the Key Information:
- Given that [tex]\(\tan x = -\frac{12}{5}\)[/tex].
- Since [tex]\(\tan x\)[/tex] is negative and [tex]\(x\)[/tex] is in the second quadrant, we know that the sine of [tex]\(x\)[/tex] is positive [tex]\((\sin x > 0)\)[/tex] and the cosine of [tex]\(x\)[/tex] is negative [tex]\((\cos x < 0)\)[/tex].
2. Construct the Right Triangle:
- In the context of a right triangle, the tangent function is defined as:
[tex]\[
\tan x = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, the opposite side is 12 and the adjacent side is 5. Note that the adjacent side will be negative in the second quadrant.
3. Calculate the Hypotenuse Using the Pythagorean Theorem:
[tex]\[
\text{hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13
\][/tex]
4. Place the Triangle on the Sketch:
- In the second quadrant, the x-coordinate (adjacent side) will be negative.
- Draw a triangle with the opposite [tex]\((y)\ coordinate being +12 and adjacent \((x)\ coordinate being -5.
### Part b) Determine the exact value for \(\frac{\sin x}{\cos 2x}\)[/tex]:
1. Calculate [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex]:
[tex]\[
\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13} = 0.9230769230769231
\][/tex]
[tex]\[
\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-5}{13} = -0.38461538461538464
\][/tex]
2. Calculate [tex]\(\cos 2x\)[/tex] Using the Double Angle Formula:
The double angle formula for cosine is:
[tex]\[
\cos 2x = 2\cos^2 x - 1
\][/tex]
First, calculate [tex]\(\cos^2 x\)[/tex]:
[tex]\[
\cos^2 x = \left(-0.38461538461538464\right)^2 = 0.14792899408284024
\][/tex]
Then, plug this into the double angle formula:
[tex]\[
\cos 2x = 2 \times 0.14792899408284024 - 1 = 0.2958579881656805 - 1 = -0.7041420118343196
\][/tex]
3. Calculate [tex]\(\frac{\sin x}{\cos 2x}\)[/tex]:
[tex]\[
\frac{\sin x}{\cos 2x} = \frac{0.9230769230769231}{-0.7041420118343196} = -1.3109243697478992
\][/tex]
### Final Answer:
[tex]\[
\frac{\sin x}{\cos 2x} = -1.3109243697478992
\][/tex]
Therefore, we've determined the exact value for [tex]\(\frac{\sin x}{\cos 2x}\)[/tex], and through a systematic breakdown, identified the relations, and crafted a neat sketch of the angle in the second quadrant.
