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Chapter 9: Problem 7

If \(\sin \phi<0\) and \(\cos \phi>0\), state the quadrant in which \(\phi\) lies.

Short Answer

Expert verified
Answer: The angle \(\phi\) lies in the fourth quadrant.

Step by step solution

01

Recall sine and cosine in quadrants

In the first quadrant (0 to 90 degrees or 0 to \(\frac{\pi}{2}\) radians), both sine and cosine are positive. In the second quadrant (90 to 180 degrees or \(\frac{\pi}{2}\) to \(\pi\) radians), sine is positive and cosine is negative. In the third quadrant (180 to 270 degrees or \(\pi\) to \(\frac{3\pi}{2}\) radians), both sine and cosine are negative. Finally, in the fourth quadrant (270 to 360 degrees or \(\frac{3\pi}{2}\) to \(2\pi\) radians), sine is negative and cosine is positive.
02

Find the quadrant

We are given that \(\sin\phi<0\) and \(\cos\phi>0\). Comparing this information with what we recall about the signs of sine and cosine in each quadrant, we find that this condition holds true in the fourth quadrant. Therefore, the angle \(\phi\) lies in the fourth quadrant.

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