Finding the Exact Value of Cos 120°

The question asks for the exact value of $\cos 120^\circ$. To find the cosine of an angle greater than $90^\circ$, we can use the concept of reference angles or the unit circle.

Understanding Angles in Different Quadrants

$120^\circ$ is an angle in the second quadrant. The quadrants are defined as follows:

• Quadrant I: $0^\circ < \theta < 90^\circ$ (All trigonometric functions are positive)
• Quadrant II: $90^\circ < \theta < 180^\circ$ (Sine is positive, Cosine and Tangent are negative)
• Quadrant III: $180^\circ < \theta < 270^\circ$ (Tangent is positive, Sine and Cosine are negative)
• Quadrant IV: $270^\circ < \theta < 360^\circ$ (Cosine is positive, Sine and Tangent are negative)

Since $120^\circ$ is in the second quadrant, we know that its cosine value will be negative.

Method 1: Using Reference Angles for Cos 120°

A reference angle is the acute angle between the terminal side of the given angle and the x-axis. For an angle $\theta$ in the second quadrant, the reference angle ($\theta'$) is calculated as $180^\circ - \theta$.

1. Find the reference angle for $120^\circ$:

   Reference angle = $180^\circ - 120^\circ = 60^\circ$.

2. Determine the sign of cosine in the second quadrant:

   In the second quadrant, the cosine function is negative.

3. Relate the cosine of the angle to the cosine of its reference angle:

   $\cos 120^\circ = -\cos (\text{reference angle})$

   $\cos 120^\circ = -\cos 60^\circ$

4. Substitute the known value of $\cos 60^\circ$:

   We know that $\cos 60^\circ = \frac{1}{2}$.

   So, $\cos 120^\circ = -\frac{1}{2}$.

Method 2: Using the Unit Circle for Cos 120°

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. For any angle $\theta$, the coordinates of the point where the terminal side of the angle intersects the unit circle are $(\cos \theta, \sin \theta)$.

1. Locate $120^\circ$ on the unit circle. This angle is in the second quadrant.
2. Consider the angle $60^\circ$ in the first quadrant. The point on the unit circle for $60^\circ$ has coordinates $(\cos 60^\circ, \sin 60^\circ)$.
3. We know $\cos 60^\circ = \frac{1}{2}$ and $\sin 60^\circ = \frac{\sqrt{3}}{2}$. So the point is $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.
4. The angle $120^\circ$ is a reflection of $60^\circ$ across the y-axis. Reflecting a point $(x, y)$ across the y-axis results in the point $(-x, y)$.
5. Therefore, the coordinates of the point for $120^\circ$ are $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$.
6. Since the x-coordinate on the unit circle represents the cosine of the angle, we have $\cos 120^\circ = -\frac{1}{2}$.

Calculating the Decimal Value

The exact value we found is $-\frac{1}{2}$. Converting this to a decimal gives:

$\cos 120^\circ = -\frac{1}{2} = -0.5$.

Comparing with Options

Let's look at the given options:

• Option 1: -0.5
• Option 2: 0
• Option 3: 0.5
• Option 4: 1

Our calculated value, $-0.5$, matches Option 1.

Conclusion

The exact value of $\cos 120^\circ$ is $-\frac{1}{2}$ or $-0.5$. This is derived using properties of angles in the second quadrant, reference angles, or the unit circle definition of cosine.

Revision Table: Common Trigonometric Values

Additional Information: Unit Circle and Reference Angles

The unit circle is a powerful tool for understanding trigonometric functions for any angle. The x-coordinate of the point where the angle's terminal side meets the circle is always $\cos \theta$, and the y-coordinate is always $\sin \theta$. This relationship helps visualize the signs of trigonometric functions in different quadrants.

Reference angles simplify finding trigonometric values for angles outside the first quadrant. By finding the value for the acute reference angle and applying the correct sign based on the quadrant, you can determine the trigonometric value for the original angle. The reference angle is always positive and between $0^\circ$ and $90^\circ$.

Key formulas for reference angles ($\theta'$) based on the angle $\theta$ (where $0^\circ < \theta < 360^\circ$):

• Quadrant I ($0^\circ < \theta < 90^\circ$): $\theta' = \theta$
• Quadrant II ($90^\circ < \theta < 180^\circ$): $\theta' = 180^\circ - \theta$
• Quadrant III ($180^\circ < \theta < 270^\circ$): $\theta' = \theta - 180^\circ$
• Quadrant IV ($270^\circ < \theta < 360^\circ$): $\theta' = 360^\circ - \theta$

Remember to check the sign of the trigonometric function in the original angle's quadrant.