George Savage
05/31/2024 · Junior High School
(iii) \( \tan \theta=\frac{3}{4}, \theta \) in quadrant III
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Tutor-Verified Answer
Step-by-step Solution
To find the values of \(\sin \theta\) and \(\cos \theta\) given that \(\tan \theta = \frac{3}{4}\) and \(\theta\) is in the third quadrant, we can use the definitions of the trigonometric functions and the Pythagorean identity.
1. **Understanding Tangent**:
\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}
\]
This means we can consider a right triangle where the opposite side is 3 and the adjacent side is 4.
2. **Finding the Hypotenuse**:
We can use the Pythagorean theorem to find the hypotenuse \(r\):
\[
r = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
3. **Finding Sine and Cosine**:
Now we can find \(\sin \theta\) and \(\cos \theta\):
\[
\sin \theta = \frac{\text{opposite}}{r} = \frac{3}{5}
\]
\[
\cos \theta = \frac{\text{adjacent}}{r} = \frac{4}{5}
\]
4. **Adjusting for Quadrant III**:
In the third quadrant, both sine and cosine are negative. Therefore:
\[
\sin \theta = -\frac{3}{5}
\]
\[
\cos \theta = -\frac{4}{5}
\]
Thus, the final values are:
\[
\sin \theta = -\frac{3}{5}, \quad \cos \theta = -\frac{4}{5}
\]
Quick Answer
\[
\sin \theta = -\frac{3}{5}, \quad \cos \theta = -\frac{4}{5}
\]
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