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 Fourth Quadrant.

Step-by-step solution

Understand the Unit Circle

Recall that the unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define the sine and cosine of an angle. The x-coordinate of a point on the unit circle is \(\cos \theta\) and the y-coordinate is \(\sin \theta\).

Identify Intervals for Sine and Cosine

Understand that \(\sin \theta < 0\) means the y-coordinate is negative, and \(\cos \theta > 0\) means the x-coordinate is positive.

Determine Quadrant Based on Conditions

Based on the unit circle, angles where the sine value is negative and the cosine value is positive lie in the Fourth Quadrant.

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the properties of angles and their relationships in a coordinate plane. The main trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each of these functions relates an angle to the ratios of sides in a right triangle.

For instance, \(\text{sin} \theta\) represents the y-coordinate of the angle \(\theta\) on the unit circle, while \(\text{cos} \theta\) is the x-coordinate. This helps us understand the position and behavior of angles within different quadrants of the unit circle.

These functions also have specific values for special angles like 0°, 30°, 45°, 60°, and 90°. It’s crucial to understand these trigonometric values to solve various problems related to angles and triangles.

Knowing the signs of these functions in different quadrants is also essential. This helps in identifying the correct quadrant for a given angle based on its sin and cos values.

Quadrants

The coordinate plane is divided into four quadrants by the x-axis and y-axis. Each quadrant represents a specific combination of positive and negative values for x and y coordinates:

• First Quadrant (Quadrant I): Both x (cosine) and y (sine) coordinates are positive.
• Second Quadrant (Quadrant II): x (cosine) is negative, y (sine) is positive.
• Third Quadrant (Quadrant III): Both x (cosine) and y (sine) coordinates are negative.
• Fourth Quadrant (Quadrant IV): x (cosine) is positive, y (sine) is negative.

Understanding which quadrant an angle lies in is crucial. In our exercise, where \(\text{sin} \theta < 0\) and \(\text{cos} \theta > 0\), we see that these conditions are satisfied in Quadrant IV, where the cosine is positive and sine is negative.

Learning the quadrant system helps in further understanding angles, their positions, and the values of their trigonometric functions.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a fundamental concept in mathematics, used to plot points, lines, and curves. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at the origin (0, 0). Points on this plane are determined by their x (horizontal) and y (vertical) coordinates.

The unit circle is a crucial tool within the coordinate plane. With a radius of 1, it is centered at the origin. The unit circle is used to understand trigonometric functions more deeply. Every point on the unit circle corresponds to an angle \(\theta\), with coordinates \(\text{cos} \theta\) and \(\text{sin} \theta\).

Using the coordinate plane and unit circle together, we can identify and analyze properties of angles and their trigonometric functions, enhancing our comprehension of their behavior and relationships.