To find the values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] given that [tex]$\tan \theta = -\frac{3}{4}$[/tex] and [tex]$\cos \theta < 0$[/tex], we can follow these steps:
1. Understand the given information:
- We know that [tex]$\tan \theta = \frac{\sin \theta}{\cos \theta}$[/tex].
- Therefore, we are given that [tex]$\frac{\sin \theta}{\cos \theta} = -\frac{3}{4}$[/tex].
2. Express [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex] in terms of a common variable:
- Let [tex]$\sin \theta = -3k$[/tex] (since the ratio of [tex]$\sin \theta$[/tex] to [tex]$\cos \theta$[/tex] is [tex]$-\frac{3}{4}$[/tex]).
- Let [tex]$\cos \theta = 4k$[/tex].
3. Use the Pythagorean identity:
- Recall that for any angle [tex]$\theta$[/tex], [tex]$\sin^2 \theta + \cos^2 \theta = 1$[/tex].
- Substituting our expressions for [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex], we get:
[tex]\[
(-3k)^2 + (4k)^2 = 1
\][/tex]
Simplify the equation:
[tex]\[
9k^2 + 16k^2 = 1
\][/tex]
[tex]\[
25k^2 = 1
\][/tex]
[tex]\[
k^2 = \frac{1}{25}
\][/tex]
[tex]\[
k = \pm \frac{1}{5}
\][/tex]
4. Determine the correct value of [tex]\( k \)[/tex]:
- Since [tex]$\cos \theta < 0$[/tex], and [tex]$\cos \theta = 4k$[/tex], [tex]\( k \)[/tex] must be negative.
- Therefore, [tex]\( k = -\frac{1}{5} \)[/tex].
5. Find the values of [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex]:
- Substitute [tex]\( k \)[/tex] back into our expressions for [tex]$\sin \theta$[/tex] and [tex]$\cos \theta$[/tex]:
[tex]\[
\sin \theta = -3k = -3 \left(-\frac{1}{5}\right) = \frac{3}{5} = 0.6
\][/tex]
[tex]\[
\cos \theta = 4k = 4 \left(-\frac{1}{5}\right) = -\frac{4}{5} = -0.8
\][/tex]
Therefore, the values of the trigonometric functions are:
[tex]\[
\sin \theta = 0.6
\][/tex]
[tex]\[
\cos \theta = -0.8
\][/tex]
