Question

If \(\theta\) is an angle in standard position, state in what quadrants its terminal side can lie if $$\theta=-47^{\circ}.$$

Short answer

The terminal side of the angle \(\theta = -47^\circ\) lies in the fourth quadrant.

Step-by-step solution

Understand the Standard Position of an Angle

An angle is in standard position if its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. Rotation from the initial side to the terminal side is counterclockwise for positive angles and clockwise for negative angles.

Determine the Direction of Rotation for Negative Angles

Since the given angle \(\theta = -47^\circ\) is negative, it means the rotation is clockwise from the positive x-axis.

Identify the Quadrant Where the Terminal Side Lies

A clockwise rotation of \(47^\circ\) from the positive x-axis will place the terminal side of the angle in the fourth quadrant because a rotation less than \(90^\circ\) will not reach the third quadrant.

Key concepts

Coordinate Plane

Imagine the coordinate plane as a map that helps us locate points in two dimensions. It is formed by two perpendicular number lines intersecting at a point called the origin. These lines are known as axes, with the horizontal axis being the x-axis and the vertical axis being the y-axis. The plane is divided into four sections, called quadrants, each representing a unique combination of positive and negative values of x and y.

Starting from the positive x-axis and moving counter-clockwise, the first quadrant comprises of both positive x and y values. As we continue to move in this direction, the second quadrant has negative x but positive y values, the third quadrant has both negative x and y values, and the fourth quadrant has positive x but negative y values. This setup acts as the stage on which angles in standard position are placed and evaluated.

Terminal Side of an Angle

When discussing angles, particularly those in standard position, the concept of the 'terminal side' becomes crucial. The terminal side is essentially the final position of a ray after it has been rotated around its endpoint, which is at the origin. For angles in standard position, this endpoint is where the x-axis and y-axis cross, known as the origin (0,0), and the initial side of the angle always aligns with the positive x-axis.

Depending on the degree measure of the angle and whether it's positive or negative, the terminal side can rotate counter-clockwise or clockwise, respectively. A negative angle, such as \(\theta = -47^\circ\), means we rotate clockwise from the positive x-axis, stopping short of a complete quadrant turn. Hence, the terminal side will be within the bounds of the quadrant that corresponds to this range of motion.

Quadrants of Angles

The quadrants of angles refer to the four different sections of the coordinate plane in which the terminal side of an angle can lie. These quadrants are numbered one through four and each gives us information about the signs of the sine, cosine, and tangent functions associated with angles located there. For instance:

• In the first quadrant, all trigonometric functions are positive.
• In the second quadrant, sine is positive, but cosine and tangent are negative.
• The third quadrant has a positive tangent, with sine and cosine being negative.
• And in the fourth quadrant, where we come back full circle for our example angle \(\theta = -47^\circ\), cosine is positive, but sine and tangent are negative.

An understanding of these quadrants helps us to predict the sign and potential value of trigonometric functions for any given angle, enhancing our analytical abilities in mathematics and related fields.