Problem 26 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \cos (x-\pi)=-\cos x $$

Step-by-Step Solution

Answer

The identity \( \cos (x-\pi)=-\cos x \) is proven by using the cosine angle subtraction formula.

Step 1: Understanding the Identity to Prove

We need to prove the trigonometric identity \( \cos (x - \pi) = -\cos x \). This is a common identity involving the cosine function and shifting by \( \pi \).

Step 2: Use the Cosine Angle Subtraction Formula

The formula for the cosine of a difference of two angles is: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] Using \( a = x \) and \( b = \pi \), we get: \[ \cos(x - \pi) = \cos x \cos(\pi) + \sin x \sin(\pi) \]

Step 3: Substitute Known Values for Cosine and Sine of Pi

From trigonometric values, we know that \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). Substitute these into the equation from Step 2: \[ \cos(x - \pi) = \cos x (-1) + \sin x (0) \] Thus, we get: \[ \cos(x - \pi) = -\cos x \]

Step 4: Conclusion

We have shown using the cosine angle subtraction formula and known trigonometric values that \( \cos(x - \pi) = -\cos x \). This completes the proof of the identity. The steps confirm that the original identity holds true.

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry. It is usually represented as \( \cos \theta \), where \( \theta \) is the angle in consideration. Cosine calculates the ratio between the length of the adjacent side and the hypotenuse in a right-angled triangle. Beyond triangles, the cosine function appears frequently in wave physics, signal processing, and even in economic modeling.

Cosine can be visualized on the unit circle, a circle with a radius of one unit centered at the origin. At any angle \( \theta \), the x-coordinate of the point where the terminal side of the angle intersects the unit circle is \( \cos \theta \). This means that cosine varies between -1 and 1 as \( \theta \) ranges from 0 to 2\( \pi \).

• Cosine is an even function: \( \cos(-\theta) = \cos \theta \).
• Important phase shifts such as \( \cos(\pi + x) = -\cos x \) occur due to the periodic nature of the cosine.

Understanding cosine and its properties is essential for solving trigonometric equations and proving identities.

Angle Subtraction Formula

The angle subtraction formula is a key tool in trigonometry. It allows us to calculate the cosine (and other trigonometric functions) of the difference between two angles. The angle subtraction formula for cosine is given by:

\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]

This formula can be used to determine the cosine of complex angles using simpler values. It’s especially useful in verifying and proving trigonometric identities, like the one we have here, \( \cos(x - \pi) = -\cos x \).

To apply this formula:

• Identify \( a \) and \( b \) as the two angles you are working with.
• Substitute known values of \( \cos a \), \( \cos b \), \( \sin a \), and \( \sin b \) as needed.
• Simplify the expression to solve for \( \cos(a - b) \).

Understanding and mastering the angle subtraction formula is powerful for solving trigonometric equations involving sums and differences of angles.

Trigonometric Values

Knowing the trigonometric values of important angles is crucial for solving almost all problems involving trigonometry. At angles like \( 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \), the values of sine and cosine have exact and easily memorizable values.

For the angle \( \pi \):

• \( \cos(\pi) = -1 \)
• \( \sin(\pi) = 0 \)

These values simplify calculations as seen in the problem \( \cos(x - \pi) = -\cos x \), where replacing the trigonometric values leads directly to the solution.

Memorizing these key trigonometric values makes computations quicker and easier. It allows students to derive proofs and solve equations without relying heavily on calculators or complex transformations. Remember that these values are derived from the unit circle's symmetry and periodic properties, reflecting how angles map onto simple trigonometric values.