Question

Name the quadrant in which the angle \(\theta\) lies. $$ \sin \theta>0, \quad \cos \theta<0 $$

Short answer

The angle \(\theta\) lies in the second quadrant.

Step-by-step solution

Analyze the sine condition

The condition \(\text{sin} \theta > 0\) tells us that the angle \(\theta\) must be located either in the first or the second quadrant, where the sine function is positive.

Analyze the cosine condition

The condition \(\text{cos} \theta < 0\) indicates that the angle \(\theta\) lies in either the second or third quadrant, where the cosine function is negative.

Determine the common quadrant

The angle \(\theta\) must satisfy both conditions: \(\text{sin} \theta > 0\) and \(\text{cos} \theta < 0\). The only quadrant where both conditions are met is the second quadrant.

Key concepts

sine function

The sine function, often written as \(\text{sin} \theta\), is a fundamental trigonometric function. It relates the angle \( \theta \) in a right triangle to the ratio of the length of the side opposite to the angle to the hypotenuse. The equation for the sine function is:
\[\text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}}\]
The sine of an angle can take values between -1 and 1. This is because the opposite side of a triangle cannot be longer than the hypotenuse. Sine values are positive in the first and second quadrants, showing the vertical height above the x-axis in these regions.
Understanding the sine function helps in determining the correct quadrant where an angle lies based on the equation given. In this exercise, \(\text{sin} \theta > 0\) tells us that \(\theta\) must be in either the first or second quadrant.

cosine function

The cosine function, written as \(\text{cos} \theta\), is another essential trigonometric function. It describes the ratio of the length of the adjacent side of the angle to the hypotenuse in a right-angled triangle. The equation for the cosine function is:
\[\text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\]
Like the sine, the cosine values range from -1 to 1. The cosine function is positive in the first and fourth quadrants because in these areas, the adjacent side relative to \(\theta\) extends to the right of the origin. Conversely, \(\text{cos} \theta < 0\) in the second and third quadrants, meaning that the adjacent side extends to the left.
For the exercise given, \(\text{cos} \theta < 0\) indicates that \(\theta\) lies in either the second or third quadrant. This information helps narrow down our options when combined with the sine function.

quadrants

The coordinate system is divided into four quadrants, each representing a unique combination of positive and negative values for sine and cosine functions. These quadrants are:

• First Quadrant (0° to 90°): \(\text{sin} \theta > 0\) and \(\text{cos} \theta > 0\).


• Second Quadrant (90° to 180°): \(\text{sin} \theta > 0\) and \(\text{cos} \theta < 0\).


• Third Quadrant (180° to 270°): \(\text{sin} \theta < 0\) and \(\text{cos} \theta < 0\).


• Fourth Quadrant (270° to 360°): \(\text{sin} \theta < 0\) and \(\text{cos} \theta > 0\).

By knowing the properties of these quadrants, we can solve trigonometric problems more easily. In this specific exercise, \(\text{sin} \theta > 0\) and \(\text{cos} \theta < 0\) both apply to the second quadrant. Therefore, the angle \(\theta\) must lie in the second quadrant.