Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

Find the values of the trigonometric functions from the given information.

Given [tex]\tan \theta = -\frac{3}{4}[/tex] and [tex]\cos \theta \ \textless \ 0[/tex], find [tex]\sin \theta[/tex] and [tex]\cos \theta[/tex].


Sagot :

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]