Question

In the third quadrant, the values of $\sin \theta$ and $\cos \theta$ are : (a) positive and negative respectively (b) negative and positive respectively (c) both positive (d) both negative

Short answer

Answer: (a) positive and negative respectively.

Step-by-step solution

Recall the definition of sine and cosine

The sine and cosine functions are defined as follows: $\sin \theta =\frac{y}{r}$ and $\cos \theta =\frac{x}{r}$, where (x, y) are the coordinates of a point on the unit circle, and r is the radius of the circle. In our case, r is always positive, and the x and y coordinates will depend on the quadrant.

Determine the signs of the x and y coordinates in the third quadrant

In the third quadrant, the x-coordinates are negative and the y-coordinates are also negative (because the points are in the lower left part of the coordinate plane). Therefore, in the third quadrant, x < 0 and y < 0.

Determine the sign of $\sin \theta$ in the third quadrant

Using the definition of the sine function, $\sin \theta =\frac{y}{r}$. Since both y and r are negative in the third quadrant, their division yields a positive result. Thus, we can conclude $\sin \theta$ is positive in the third quadrant.

Determine the sign of $\cos \theta$ in the third quadrant

Using the definition of the cosine function, $\cos \theta =\frac{x}{r}$. In the third quadrant, x is negative and r is positive. Therefore, their division yields a negative result. Hence, we can conclude that $\cos \theta$ is negative in the third quadrant.

Match the signs of $\sin \theta$ and $\cos \theta$ with the given options

Now that we know that $\sin \theta$ is positive and $\cos \theta$ is negative in the third quadrant, we can match this information with the given options. Option (a) states: positive and negative respectively. This option matches our findings. Therefore, the correct answer is (a) positive and negative respectively.

Key concepts

Sine Function

The sine function, denoted as $\sin \theta$, relates to the y-coordinate of a point on a unit circle.
It is defined as $\sin \theta =\frac{y}{r}$, where

• $y$ is the vertical coordinate,
• $r$ is the radius of the circle.

In the unit circle, $r$ is always 1, simplifying the equation to $\sin \theta =y$. This function is positive in quadrants where the y-value is positive, typically the first and second quadrants.
In the third quadrant, however, the sine function can be trickier.
Despite being in a region where both x and y values are negative, the division of
a negative $y$ by a negative $r$ results in a positive sine.
This is a key point when understanding trigonometric functions in different quadrants.

Cosine Function

The cosine function, known as $\cos \theta$, is closely related to the horizontal coordinate of a point on the unit circle.
Its definition is $\cos \theta =\frac{x}{r}$, where

• $x$ represents the horizontal coordinate,
• $r$ is the radius of the unit circle.

For a unit circle, $r$ is always 1, making $\cos \theta =x$. This allows the cosine function to depict the horizontal distance of a point from the origin.
The cosine is positive in quadrants where the x-value is positive, typically the first and fourth quadrants.
However, in the third quadrant, $x$ is negative as it lies to the left of the y-axis.
Consequently, $\cos \theta =\frac{x}{r}$ results in a negative value due to negative $x$, explaining why cosine is negative in the third quadrant.

Unit Circle

The unit circle is an essential concept in trigonometry, especially for understanding
how the sine and cosine functions work. Consider a circle with a radius of 1 centered on the origin
of a coordinate plane.
The circle divides the plane into four quadrants, each having special properties for trigonometric functions.

• The first quadrant (0° to 90°) has both x and y positive.
• The second quadrant (90° to 180°) has x negative and y positive.
• In the third quadrant (180° to 270°), both x and y are negative.
• The fourth quadrant (270° to 360°) reverses, with x positive and y negative.

Understanding these quadrant properties can help in determining the sign of$\sin \theta$ and $\cos \theta$.
For any angle $\theta$ on the unit circle, the coordinates (x, y) corresponding to that angle
tell you exactly what the sine and cosine values will be. In the context of the third quadrant, it is
crucial to note that both sine and cosine relate to negative x and y, resulting in a positive sine and negative cosine.