In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies.sin θ > 0, cos θ > 0

Verified step by step guidance

1. Step 1: Recall the unit circle and the signs of trigonometric functions in each quadrant.
2. Step 2: Understand that in the unit circle, the sine function (sin θ) represents the y-coordinate, and the cosine function (cos θ) represents the x-coordinate.
3. Step 3: Note that sin θ > 0 indicates that the y-coordinate is positive, which occurs in Quadrants I and II.
4. Step 4: Note that cos θ > 0 indicates that the x-coordinate is positive, which occurs in Quadrants I and IV.
5. Step 5: Determine the quadrant where both sin θ > 0 and cos θ > 0 are true, which is Quadrant I.

Key Concepts

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants based on the signs of the x (cosine) and y (sine) coordinates. Quadrant I has both sine and cosine positive, Quadrant II has sine positive and cosine negative, Quadrant III has both negative, and Quadrant IV has sine negative and cosine positive. Understanding these quadrants is essential for determining the location of an angle based on the signs of its trigonometric functions.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. Specifically, sine represents the ratio of the opposite side to the hypotenuse, while cosine represents the ratio of the adjacent side to the hypotenuse. The signs of these functions in different quadrants help identify the angle's location in the coordinate plane.

Standard Position of an Angle

An angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. The angle is measured counterclockwise from the initial side. This concept is crucial for determining the quadrant in which the terminal side of the angle lies, based on the values of sine and cosine.