Question

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

Short answer

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

Step-by-step solution

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.

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.

Key concepts

Sine Function

The sine function is one of the fundamental trigonometric functions and is usually written as \( \sin \theta \). It relates to the y-coordinate on the unit circle, which is based on angles measured from the positive x-axis. - When thinking about sine, visualize the height from the x-axis of the point on a unit circle.- The sine value ranges from -1 to 1.In terms of quadrants:- First Quadrant: \( \sin \theta > 0 \)- Second Quadrant: \( \sin \theta > 0 \)- Third Quadrant: \( \sin \theta < 0 \)- Fourth Quadrant: \( \sin \theta < 0 \)In the original exercise, since \( \sin \phi < 0 \), it implies it could either be in the third or fourth quadrant. But, to find the exact quadrant, we consider cosine too.

Cosine Function

The cosine function represents the x-coordinate of the point on the unit circle. It is written as \( \cos \theta \) and gives an idea of how far left or right a point is from the center of the circle.- Like sine, cosine values also range from -1 to 1.- Cosine indicates the horizontal distance in the unit circle.In each quadrant, cosine behaves as follows:- First Quadrant: \( \cos \theta > 0 \)- Second Quadrant: \( \cos \theta < 0 \)- Third Quadrant: \( \cos \theta < 0 \)- Fourth Quadrant: \( \cos \theta > 0 \)For the given problem, \( \cos \phi > 0 \), which narrows down the location of \( \phi \) to either the first or fourth quadrant. Knowing sine’s sign confirms it's in the fourth quadrant.

Trigonometric Signs

Understanding the signs of trigonometric functions in different quadrants helps determine the exact position of an angle. Memorizing these signs can be particularly useful in trigonometry problems.Each quadrant has its unique characteristics based on the signs of sine and cosine:- First Quadrant: Both \( \sin \theta \) and \( \cos \theta \) are positive.- Second Quadrant: \( \sin \theta \) is positive, \( \cos \theta \) is negative.- Third Quadrant: Both \( \sin \theta \) and \( \cos \theta \) are negative.- Fourth Quadrant: \( \sin \theta \) is negative, \( \cos \theta \) is positive.Remembering these signs lets you quickly identify which quadrant an angle lies in when you know the values of sine and cosine. For instance, if \( \sin \phi < 0 \) and \( \cos \phi > 0 \), the angle \( \phi \) must be in the fourth quadrant, which matches the solution in the exercise provided.