Problem 32 - Algebra 2

Question

Verify that each of the following is an identity. \(\cos (\pi-\theta)=-\cos \theta\)

Step-by-Step Solution

Answer

The identity \( \cos (\pi - \theta) = -\cos \theta \) is verified as true.

Step 1: Understand the Identity

The identity we need to verify is \( \cos (\pi - \theta) = -\cos \theta \). This equation claims that the cosine of \( \pi \) minus an angle \( \theta \) is equal to the negative cosine of that angle \( \theta \).

Step 2: Apply Cosine Difference Formula

The cosine of a difference formula is \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). Here, \( a = \pi \) and \( b = \theta \), so we apply the formula: \( \cos(\pi - \theta) = \cos \pi \cos \theta + \sin \pi \sin \theta \).

Step 3: Simplify Using Known Values

We know that \( \cos \pi = -1 \) and \( \sin \pi = 0 \). Substituting these values into the equation: \( \cos(\pi - \theta) = (-1) \cos \theta + 0 \cdot \sin \theta \). This simplifies to \( -\cos \theta \).

Step 4: Confirm the Identity

After simplifying, we found that \( \cos(\pi - \theta) = -\cos \theta \). This matches exactly with the identity given in the exercise, thus confirming it is true.

Key Concepts

Cosine Difference Formula

The cosine difference formula is a crucial identity in trigonometry. It helps us find the cosine of the difference between two angles. The formula is given by \( \cos(a - b) = \cos a \cos b + \sin a \sin b \). This formula is handy because it breaks down the complex measurement of angle differences into a combination of basic trigonometric functions.

• \( \cos(a) \) and \( \cos(b) \) represent the cosine of the two angles.
• \( \sin(a) \) and \( \sin(b) \) represent the sine of those angles.

When you encounter situations in math or science requiring the cosine of a difference, this formula is your go-to. It simplifies the calculations by transforming angle differences into simple operations involving products of sine and cosine. Keep in mind that with the right values and application, it can make tricky problems much easier to solve.

Angle Subtraction Identity

The angle subtraction identity is another name for the cosine difference formula. It's pivotal for solving many trigonometric equations and verifying identities. Understanding this concept helps build upon fundamental trigonometry skills.
When using the angle subtraction identity:

• Your first step is identifying the two angles \( a \) and \( b \).
• Then, substitute these angles into the formula \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
• Finally, simplify using known values of trigonometric functions for any standard angles involved, like \( \pi \) or \( \theta \).

This identity helps in breaking down complex angles into familiar components, making evaluation easier. For example, to verify \( \cos(\pi - \theta) = -\cos \theta \), using the identity makes it easy to substitute known sine and cosine values of \( \pi \) and simplify the expression.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in mathematics for modeling cyclic phenomena. Each function represents a relationship between the angles and lengths in right-angled triangles or unit circles.

• **Cosine** measures the adjacent side to the hypotenuse.
• **Sine** measures the opposite side to the hypotenuse.
• **Tangent** is the ratio of sine to cosine.

In the context of the cosine difference formula \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), we use the basic functions:

• \( \cos \pi = -1 \) and \( \sin \pi = 0 \) from known trigonometric values.

These functions allow us to evaluate complex expressions by substituting known trigonometric values for specific angles. Understanding these commonly used trigonometric functions empowers you to verify identities, solve equations, and model real-world problems.