Problem 36 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \cos (\pi-x)=-\cos x $$

Step-by-Step Solution

Answer

The given identity \(\cos(\pi -x) = -\cos x\) is verified successfully by using the cosine difference identity and applying the values of \(\cos \pi\) and \(\sin \pi\) from the unit circle.

Step 1: Understand the given identity

The given identity is \(\cos(\pi -x) = -\cos x\). The goal is to show that the left side of the equation is identical to the right side.

Step 2: Use the Cosine Difference Identity

One of the cosine difference identities states that \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). Use this identity to rewrite \(\cos(\pi -x)\). Replace \(A\) with \(\pi\) and \(B\) with \(x\). You should obtain \(\cos(\pi -x) = \cos \pi \cos x + \sin \pi \sin x \)

Step 3: Evaluate Cosine and Sine Values

Now, use the unit circle to find the values of \(\cos \pi\) and \(\sin \pi\). These are \(\cos \pi = -1\) and \(\sin \pi = 0\). Substitute these into the equation to obtain \(-1*\cos x + 0*\sin x = -\cos x\). The term with the sine disappears since it's multiplied by zero, which leaves us with \(-\cos x = -\cos x\)

Step 4: Final Verification

The final expression matches the right-hand side of the original identity, confirming that the identity is true.

Key Concepts

Cosine Difference Identity

The cosine difference identity is a trigonometric formula used to express the cosine of an angle difference in terms of the cosine and sine of the individual angles. This identity is written in its general form as: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
In the original exercise, we apply this identity to \( \cos(\pi - x) \).To properly use the identity, substitute \( A = \pi \)and \( B = x \).This substitution yields:\[ \cos(\pi - x) = \cos \pi \cos x + \sin \pi \sin x \]
Using this identity breaks down the problem into evaluating basic trigonometric functions using known angles and simplifies the expression.

Unit Circle

The unit circle is a powerful tool in trigonometry that helps us evaluate trigonometric functions for common angles. It's a circle with a radius of 1 centered at the origin of a coordinate plane.

• The angle \( \pi \) (or \( 180^\circ \) degrees) corresponds to the point \((-1, 0)\)on the unit circle.
• The cosine of an angle on the unit circle is the x-coordinate of the corresponding point.
• The sine of an angle is the y-coordinate of the point.

From this, for the angle \( \pi \):

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

These values enable us to compute expressions involving angles, such as those required for the cosine difference identity.

Evaluating Trigonometric Functions

Evaluating trigonometric functions involves finding the values of sine, cosine, and other trigonometric functions for specific angles. In the context of the original problem:

• We first identify the angle to work with, which is \( \pi - x \).
• Next, we apply relevant identities like the cosine difference identity.
• Then, we determine the cosine and sine values needed using the unit circle.

For example, by substituting the known values from the unit circle, \(-1\) for \( \cos \pi \) and \(0\) for \( \sin \pi \), we can simplify the expression: \[ \cos \pi \cos x + \sin \pi \sin x = -\cos x + 0 \cdot \sin x \]
This results in:\[-\cos x\], which matches the right side of the identity we aimed to verify, such as \( \cos(\pi - x) = -\cos x \).This confirms the identity through correct evaluation.