Problem 35 - Algebra and Trigonometry

Question

Find the quadrant in which \(\theta\) lies from the information given. $$ \sin \theta<0 \quad \text { and } \quad \cos \theta<0 $$

Step-by-Step Solution

Answer

\( \theta \) lies in the third quadrant.

Step 1: Identify Quadrant Conditions for Sine

Recall that the sine function, \( \sin \theta \), is negative in the third and fourth quadrants. Therefore, given that \( \sin \theta < 0 \), \( \theta \) must lie either in the third or fourth quadrant.

Step 2: Identify Quadrant Conditions for Cosine

Next, recall that the cosine function, \( \cos \theta \), is negative in the second and third quadrants. Therefore, given that \( \cos \theta < 0 \), \( \theta \) must lie in the second or third quadrant.

Step 3: Determine the Quadrant Satisfying Both Conditions

Now, combine the restrictions on \( \sin \theta \) and \( \cos \theta \). The only quadrant where both \( \sin \theta < 0 \) and \( \cos \theta < 0 \) are true is the third quadrant.

Key Concepts

Quadrants in Trigonometry

Understanding quadrants in trigonometry is essential when solving problems related to angles and functions. The coordinate plane is divided into four quadrants. Each has unique characteristics depending on the signs of the x (horizontal) and y (vertical) coordinates.

- **First Quadrant**: Both x and y are positive. Thus, sine and cosine are positive. - **Second Quadrant**: x is negative and y is positive. Here, sine is positive, but cosine is negative. - **Third Quadrant**: Both x and y are negative. This makes sine and cosine negative. - **Fourth Quadrant**: x is positive and y is negative. Cosine is positive while sine remains negative.

When angles are placed on this plane, these quadrants help determine the sign of trigonometric functions. For instance, if both sine and cosine are negative, as given in the exercise, the angle lies in the third quadrant. Remember, practicing identifying the quadrants will make solving similar problems much easier.

Sine Function

The sine function, denoted as \( \sin \theta \), tells us about the vertical position of a point on a circle with radius one, known as the unit circle.

- **Positive Values**: In the unit circle, sine is positive in the first and second quadrants where the y-coordinate is positive.- **Negative Values**: Sine takes negative values in the third and fourth quadrants. This is due to the y-coordinate being below the x-axis.

Understanding when sine is positive or negative helps snugly fit your angle into a quadrant. In our exercise, because \( \sin \theta < 0 \), the angle \( \theta \) can either be in the third or fourth quadrant. This assists in narrowing down options when combined with other trigonometric restrictions.

Cosine Function

The cosine function, \( \cos \theta \), is related to the horizontal position of a point on the unit circle.

- **Positive Values**: Cosine takes positive values in the first and fourth quadrants. Here, the x-coordinate is to the right of the y-axis.- **Negative Values**: In the unit circle, cosine becomes negative in the second and third quadrants where the x-coordinate falls to the left side of the y-axis.

With a grasp of when the cosine function is negative, it further helps situate the angle within specific quadrants. Given in the exercise that \( \cos \theta < 0 \), the angle \( \theta \) could lie in either the second or third quadrant.

Combining the cosine and sine conditions shows that both must be negative, placing the angle firmly in the third quadrant. This ensures that our analysis aligns correctly with trigonometric properties and coordinates.