Problem 53 - Algebra 2

Question

Use the sum and difference formulas to verify each identity. $$ \cos (\pi-\theta)=-\cos \theta $$

Step-by-Step Solution

Answer

Yes, the identity \( \cos(\pi-\theta) = -\cos \theta \) holds true due to the difference formula and the evaluative trigonometric values

Step 1: Understanding the Difference Formula

Recall the difference formula for cosine, which is: \\( \cos(a - b) = \cos a\cos b + \sin a\sin b \) \ This equation will be used to solve this problem.

Step 2: Apply the Difference Formula

Applying the difference formula for given identity:\( cos (\pi - \theta) = \cos \pi\cos \theta + \sin \pi\sin \theta \)

Step 3: Evaluate the Trigonometric Values

From the unit circle, we know that at \( \pi \), \( \cos \pi = -1 \) and \( \sin \pi = 0 \). Substituting these values, we get:\( \cos (\pi -\theta) = -1\cdot\cos \theta + 0\cdot\sin \theta = -\cos \theta \)

Key Concepts

Cosine Difference Identity

The cosine difference identity is a very useful tool in trigonometry. It allows us to express the cosine of a difference of two angles in terms of the cosines and sines of the individual angles. The formula is given by:

• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

This identity helps decompose complex trigonometric expressions into simpler parts. In our exercise, we applied this identity to \( \cos(\pi - \theta) \). By substituting \( a = \pi \) and \( b = \theta \), we could break it down using the known values of cosine and sine at \( \pi \). This type of manipulation is at the core of solving trigonometric problems using identities, aiding in both simplification and verification of expressions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables present, as long as they are within the domain of the trigonometric functions involved. These identities are extremely powerful in simplifying and verifying complex trigonometric expressions. Some of the most common types of trigonometric identities include the basic Pythagorean identities, reciprocal identities, and co-function identities.
In our exercise, we use one specific identity—the cosine difference identity—to prove that \( \cos(\pi - \theta) = -\cos \theta \). By understanding and using identities, we not only simplify expressions but also solve equations and prove new relationships.
Studying these identities helps establish a strong foundation in trigonometry, paving the way for deeper exploration into analytical and applied mathematics.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It plays a central role in trigonometry because it helps us define sine and cosine for all angles. Every point on the unit circle corresponds to an angle, where the x-coordinate gives the cosine value and the y-coordinate gives the sine value of that angle.
In the exercise, recognizing the values from the unit circle was crucial. At angle \( \pi \), the unit circle tells us that \( \cos \pi = -1 \) and \( \sin \pi = 0 \). These values are essential to evaluating the trigonometric expressions and applying the cosine difference identity effectively. Reviewing the unit circle helps to quickly access these values, thus facilitating quicker problem-solving.

Evaluating Trigonometric Functions

Evaluating trigonometric functions, such as cosine and sine, involves determining their values based on given angles or expressions. This often requires using known values from the unit circle or applying trigonometric identities.
In the given problem, once we applied the cosine difference identity, we needed to evaluate \( \cos \pi \) and \( \sin \pi \), which are simple thanks to the unit circle. By substituting these values into our identity, we could verify that \( \cos(\pi - \theta) = -\cos \theta \). Being able to evaluate these functions accurately helps in solving trigonometric equations and validating different identities, which are common tasks in both pure and applied mathematics.