Solve: Cos Θ = -1

Let's dive into the world of trigonometry to answer the question: For which angle $\theta$ is $\cos \theta = -1$? This might seem like a tricky question at first glance, but with a little understanding of the cosine function and the unit circle, it becomes quite straightforward. We'll explore the behavior of cosine, visualize it on the unit circle, and then pinpoint the exact angle that satisfies this condition.

Understanding the Cosine Function and the Unit Circle

The cosine function, denoted as $\cos \theta$, is a fundamental part of trigonometry. It's intimately related to the unit circle, which is a circle with a radius of 1 centered at the origin (0,0) on a Cartesian coordinate system. When we talk about an angle $\theta$ in trigonometry, we often imagine it in standard position, with its vertex at the origin and its initial side lying along the positive x-axis. The terminal side of the angle is then a ray emanating from the origin. Where this terminal side intersects the unit circle, we get a point (x, y). The cosine of the angle $\theta$ is defined as the x-coordinate of this point, so $\cos \theta = x$. The sine of the angle is the y-coordinate, $\sin \theta = y$.

Now, let's consider the possible values for the x-coordinate on the unit circle. Since the circle has a radius of 1, the x-values can range from -1 (at the leftmost point of the circle) to +1 (at the rightmost point of the circle). The cosine function, representing this x-coordinate, therefore has a range of [-1, 1]. We are looking for the angle $\theta$ where the x-coordinate on the unit circle is exactly -1. This occurs at the point on the unit circle that is furthest to the left.

Pinpointing the Angle

Visualize the unit circle. As we rotate a ray from the positive x-axis counterclockwise, the x-coordinate of the intersection point changes. When the angle is $0^{\circ}$ (or 0 radians), the terminal side lies along the positive x-axis, and the intersection point is (1, 0). Here, $\cos 0^{\circ} = 1$. As we increase the angle towards $90^{\circ}$ (or $\pi/2$ radians), the terminal side moves into the first quadrant, and the x-coordinate decreases, becoming 0 at $90^{\circ}$ (point (0, 1)). As we continue to $180^{\circ}$ (or $\pi$ radians), the terminal side lies along the negative x-axis. At this point, the intersection is (-1, 0). This is precisely where the x-coordinate is -1! Therefore, $\cos 180^{\circ} = -1$.

It's crucial to remember that trigonometric functions are periodic. This means they repeat their values at regular intervals. The cosine function has a period of $360^{\circ}$ (or $2\pi$ radians). So, if $\cos \theta = -1$ at $\theta = 180^{\circ}$, it will also be -1 at $180^{\circ} + 360^{\circ}$, $180^{\circ} + 2 \times 360^{\circ}$, and so on. In general, $\cos \theta = -1$ for all angles of the form $\theta = 180^{\circ} + n \times 360^{\circ}$, where 'n' is any integer (..., -2, -1, 0, 1, 2, ...).

Evaluating the Options

Now, let's look at the given options to see which one fits this condition:

• A. $540^{\circ}$: To check this, we can subtract multiples of $360^{\circ}$ to find a coterminal angle within $0^{\circ}$ to $360^{\circ}$. $540^{\circ} - 360^{\circ} = 180^{\circ}$. Since $\cos 180^{\circ} = -1$, $540^{\circ}$ is a valid angle for which $\cos \theta = -1$. Indeed, $540^{\circ} = 180^{\circ} + 1 imes 360^{\circ}$.
• B. $360^{\circ}$: The angle $360^{\circ}$ is a full rotation. The terminal side ends up on the positive x-axis, just like $0^{\circ}$. So, $\cos 360^{\circ} = 1$. This is not -1.
• C. $270^{\circ}$: At $270^{\circ}$, the terminal side lies along the negative y-axis, at the point (0, -1). The x-coordinate is 0, so $\cos 270^{\circ} = 0$. This is not -1.
• D. $450^{\circ}$: Similar to option A, let's find a coterminal angle. $450^{\circ} - 360^{\circ} = 90^{\circ}$. We know that $\cos 90^{\circ} = 0$. This is not -1.

Conclusion

Based on our analysis, the angle for which $\cos \theta = -1$ is $180^{\circ}$ and any angle that is coterminal with it. Among the given options, A. $540^{\circ}$ is the correct answer because $540^{\circ}$ is coterminal with $180^{\circ}$. It represents one full rotation ($360^{\circ}$) plus an additional $180^{\circ}$.

For further exploration into trigonometric functions and their properties, you can visit resources like Khan Academy or Wolfram MathWorld.