Problem 55 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\cos (\pi-x)=-\cos x$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity \(\cos (\pi-x)=-\cos x\) using a geometric approach. Answer: By analyzing the positions of angles x and \((\pi-x)\) on the unit circle, we find that the x-coordinate of the terminal point for the angle \((\pi-x)\) is the negative of the x-coordinate for the angle x, since they are in opposite quadrants. This demonstrates that the cosine values are related by a negative sign, proving the trigonometric identity \(\cos (\pi-x)=-\cos x\).

Step 1: Identify the given trigonometric identity

We are given the identity to prove: $$\cos (\pi-x)=-\cos x$$

Step 2: Recall the definition of cosine in the unit circle

In the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle. For example, if we have an angle \(x\) in radians, we can find its terminal point (x, y) and the cosine function is given by the x-coordinate of that point: \(\cos(x) = x.\)

Step 3: Visualize the angles x and \((\pi-x)\) on the unit circle

Consider x to be a positive angle in the first or second quadrant. Then, \(\pi-x\) can be seen as the angle formed by rotating π radians counterclockwise (or x radians clockwise) from the terminal side of x. This will result in the corresponding terminal side being in the third or fourth quadrant.

Step 4: Determine the cosine values of both angles

Since the angle \((\pi-x)\) is in the third or fourth quadrant, its cosine value will be negative (recall that cosine values are negative for angles in the second and third quadrants). Meanwhile, the angle x is in the first or second quadrant, meaning its cosine value will be positive.

Step 5: Prove the equality

Now we can see that the x-coordinate of the terminal point for the angle \((\pi-x)\) is the negative of the x-coordinate for the angle x, since they are in opposite quadrants. Therefore, the two cosine values are related by a negative sign, which proves the given identity: $$\cos (\pi-x)=-\cos x$$

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that aids in understanding the relationships between angles and trigonometric functions. This circle has a radius of 1 and is centered at the origin of a coordinate system.

Here are some essential aspects of the unit circle:

• Every point on the unit circle has coordinates that are determined by the angle \( heta \) measured from the positive x-axis.
• The x-coordinate of a point on the circle is equal to \( \cos(\theta) \) and the y-coordinate is equal to \( \sin(\theta) \).
• Since the radius is 1, the equation of the unit circle is \( x^2 + y^2 = 1 \).
• The unit circle helps to visualize trigonometric functions and understand their periodic nature as you rotate around the circle.

Understanding the unit circle's structure allows you to better comprehend how the \( \cos(\theta) \) and \( \sin(\theta) \) values change depending on the angle \( \theta \). It also provides insight into why certain identities, like the one in the exercise, hold true by leveraging symmetry properties.

Cosine Function

The cosine function is a fundamental trigonometric function that forms the backbone of many mathematical applications.

Let's dive deeper into some key properties of the cosine function:

• The cosine function relates an angle to the x-coordinate on the unit circle. For any angle \( \theta \), \( \cos(\theta) \) is the value of the x-coordinate where the terminal side of the angle (radius) intersects the unit circle.
• Cosine is a periodic function with a period of \( 2\pi \). This means that after every \( 2\pi \) radians, the value of \( \cos(\theta) \) repeats.
• The cosine function has even symmetry, which implies that \( \cos(\theta) = \cos(-\theta) \).
• Values of \( \cos(\theta) \) range from -1 to 1 for angles between 0 and \( \pi \).

Understanding these properties of the cosine function is crucial for analyzing trigonometric identities and transformations, like proving the identity \( \cos(\pi-x) = -\cos(x) \) by using properties of angles and the cosine function in different quadrants.

Angle Quadrants

The coordinate plane is divided into four quadrants, and each quadrant has distinct characteristics that affect the sign and value of trigonometric functions, such as cosine.

Here's a brief overview of angle quadrants:

• Quadrant I: Angles range from \( 0 \) to \( \frac{\pi}{2} \) radians. Here, both sine and cosine values are positive.
• Quadrant II: Angles range from \( \frac{\pi}{2} \) to \( \pi \) radians. The sine is positive while the cosine is negative.
• Quadrant III: Angles range from \( \pi \) to \( \frac{3\pi}{2} \) radians. Both sine and cosine values are negative.
• Quadrant IV: Angles range from \( \frac{3\pi}{2} \) to \( 2\pi \). Here, sine is negative, and cosine is positive.

Knowing the quadrant where an angle lies helps determine the signs of sine and cosine values, based on the knowledge that these functions switch signs as the angle moves from one quadrant to the next. For the identity in the exercise, the angles \( x \) and \( \pi-x \) are in opposite quadrants, explaining why their cosine values have opposite signs.