To find the value of [tex]\(\cos \theta\)[/tex] given that [tex]\(\sin \theta = -\frac{84}{85}\)[/tex] and the angle [tex]\(\theta\)[/tex] terminates in quadrant III, follow these steps:
1. Identify Quadrant III Characteristics:
- In quadrant III, both sine ([tex]\(\sin \theta\)[/tex]) and cosine ([tex]\(\cos \theta\)[/tex]) are negative.
2. Use the Pythagorean Identity:
- The Pythagorean identity states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Since we know [tex]\(\sin \theta = -\frac{84}{85}\)[/tex], we can substitute this into the identity.
3. Calculate [tex]\(\sin^2 \theta\)[/tex]:
- First, square the given value of [tex]\(\sin \theta\)[/tex]:
[tex]\[
\left( -\frac{84}{85} \right)^2 = \frac{84^2}{85^2} = \frac{7056}{7225}
\][/tex]
4. Find [tex]\(\cos^2 \theta\)[/tex]:
- Substitute [tex]\(\sin^2 \theta\)[/tex] into the Pythagorean identity:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{7056}{7225}
\][/tex]
- Simplify the expression:
[tex]\[
\cos^2 \theta = \frac{7225}{7225} - \frac{7056}{7225} = \frac{7225 - 7056}{7225} = \frac{169}{7225}
\][/tex]
5. Calculate [tex]\(\cos \theta\)[/tex]:
- Take the square root of both sides to solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \pm \sqrt{\frac{169}{7225}} = \pm \frac{\sqrt{169}}{\sqrt{7225}} = \pm \frac{13}{85}
\][/tex]
- Since [tex]\(\theta\)[/tex] is in quadrant III, where cosine is negative, choose the negative value:
[tex]\[
\cos \theta = -\frac{13}{85}
\][/tex]
Thus, the value of [tex]\(\cos \theta\)[/tex] is [tex]\(-\frac{13}{85}\)[/tex].
In decimal form, it is approximately:
[tex]\[
\cos \theta \approx -0.1529
\][/tex]
Therefore, [tex]\(\cos \theta = -\frac{13}{85}\)[/tex] and approximately [tex]\(-0.1529\)[/tex].
