Problem 56 - Algebra 2

Question

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

Step-by-Step Solution

Answer

-\cos(\theta) verifies the identity. Indeed, cos(\pi+\theta) equals -\cos theta.

Step 1: Understand the formula

The formula for cos(x+y) states: \(\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)\). This will provide the foundation in solving the problem. We will use this formula with \(x = \pi\) and \(y = \theta\).

Step 2: Substitution

\(\cos(\pi + \theta) = \cos(\pi)\cos(\theta) - \sin(\pi)\sin(\theta)\). Now we substitute the values of \(cos(\pi)\) and \(sin(\pi)\).

Step 3: Evaluation

Given that \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\), we can substitute these values into the formula. This simplifies to: \(\cos(\pi + \theta) = -1*\cos(\theta) - 0 *\sin(\theta) = -\cos(\theta)\).

Key Concepts

Sum and Difference Formulas

The sum and difference formulas are essential tools in trigonometry. They help simplify the expressions involving the trigonometric functions sine and cosine when two angles are added or subtracted.

Here's how the formulas generally work:

• Cosine of a sum formula: \( \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) \).
• Cosine of a difference formula: \( \cos(x - y) = \cos(x)\cos(y) + \sin(x)\sin(y) \).
• Sine of a sum formula: \( \sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y) \).
• Sine of a difference formula: \( \sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y) \).

These identities allow you to express the trigonometric functions of sum or difference of angles in terms of the functions of individual angles. Using these formulas can make complex problems more manageable and help in verifying identities like the one given.

When approaching a problem, it's important to identify the angles as they fit into the specific formula, facilitating substitutions and simplifications.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse.

In terms of the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side intersects the circle.

Key properties of the cosine function that can be helpful include:

• Periodic Function: Cosine has a period of \(2\pi\). After \(2\pi\), the pattern of the function repeats.
• Even Function: This means that \(\cos(-x) = \cos(x)\).
• Values at Quadrantal Angles: For example, \(\cos(0) = 1\), \(\cos(\pi/2) = 0\), \(\cos(\pi) = -1\), and \(\cos(3\pi/2) = 0\).

Understanding these characteristics helps tremendously when solving and simplifying trigonometric identities. For instance, recognizing that \(\cos(\pi) = -1\) was indispensable in simplifying the identity \(\cos(\pi + \theta) = -\cos(\theta)\).

This property aligns with how the cosine function behaves symmetrically about the y-axis.

Angle Addition Identities

Angle addition identities are specific types of formulas used to find the trigonometric function values for the sum of two angles. They are closely related to the sum and difference formulas.

The cosine of the sum identity is written as \( \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y) \). This identity is incredibly useful in verifying trigonometric expressions, just like in the original exercise where \( \cos(\pi + \theta) \) was verified as \(-\cos(\theta)\). By substituting "\(x\)" with \(\pi\) and "\(y\)" with \(\theta\), and knowing the cosine and sine values of special angles, you can simplify expressions easily.

Another aspect to remember:

• This identity showcases that trigonometric functions of compound angles can be expressed in terms of functions of simple angles.
• It teaches us how trigonometric ratios behave under the addition or subtraction of angles, an essential understanding for many areas in mathematics and physics.

These identities not only make solving trigonometric problems more structured but also allow deeper insights into the relationships between angles and their trigonometric functions.