Problem 19

Question

In Exercises $17-22$, let $\theta$ be an angle in standard position. Name the quadrant in which $\theta$ lies $$\sin \theta<0, \quad \cos \theta<0$$

Step-by-Step Solution

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Answer

The angle $ \theta $ lies in the third quadrant.

1Step 1: Understand Trigonometry in Each Quadrant

The signs of $ \sin\theta $ and $ \cos\theta $ vary in different quadrants. In the first quadrant (0 to 90 degrees), both $ \sin\theta $ and $ \cos\theta $ are positive. In the second quadrant (90 to 180 degrees), $ \sin\theta $ is positive and $ \cos\theta $ is negative. In the third quadrant (180 to 270 degrees), both $ \sin\theta $ and $ \cos\theta $ are negative. In the fourth quadrant (270 to 360 degrees), $ \sin\theta $ is negative and $ \cos\theta $ is positive.

2Step 2: Identify the Quadrant

As per the given condition, $ \sin\theta<0 $ and $ \cos\theta<0 $. So both sine and cosine are negative. Based on our understanding in Step 1, we can say that only in the third quadrant ($ \theta $ between 180 and 270 degrees), both sine and cosine of the angle are negative.

Key Concepts

Trigonometric QuadrantsSine and Cosine SignsStandard Position Angle

Trigonometric Quadrants

Understanding trigonometric quadrants is crucial in trigonometry. Imagine a circle divided into four sections, called quadrants. Each quadrant corresponds to a range of angles:

First Quadrant: 0 to 90 degrees
Second Quadrant: 90 to 180 degrees
Third Quadrant: 180 to 270 degrees
Fourth Quadrant: 270 to 360 degrees

These quadrants help us determine the signs of trigonometric functions like sine and cosine. It's like checking the weather in each region of a map. Depending on where you are, the conditions change!
Remember, angles in standard position start from the positive x-axis, creating these distinct regions as they rotate counterclockwise.

Sine and Cosine Signs

The signs of sine and cosine functions vary depending on the quadrant in which the angle lies. Let's try to understand this better:

First Quadrant: Both sine ($ \sin\theta > 0$ ) and cosine ($ \cos\theta > 0$ ) values are positive.
Second Quadrant: Sine is positive ($ \sin\theta > 0$), but cosine is negative ($ \cos\theta < 0$).
Third Quadrant: Both sine ($ \sin\theta < 0$) and cosine ($ \cos\theta < 0$ ) values are negative.
Fourth Quadrant: Sine is negative ($ \sin\theta < 0$), and cosine is positive ($ \cos\theta > 0$).

These rules make it easier to identify the quadrant of any angle just by knowing the signs of sine and cosine.
This is crucial for solving trigonometric problems effectively.

Standard Position Angle

The concept of a standard position angle is foundational in trigonometry. A standard position angle starts with its vertex at the origin of the coordinate plane.

The initial side is placed along the positive x-axis.
It measures angles by their rotation from the initial side.
Rotations are counterclockwise for positive angles and clockwise for negative angles.

Visualizing angles in this way helps in understanding their placement and relationship with trigonometric functions.
It is also helpful when plotting angles on a coordinate grid, making it easier to determine the quadrant in which an angle lies.

Problem 19

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