Problem 43 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \tan \left(\theta+\frac{\pi}{4}\right)=\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta} $$

Step-by-Step Solution

Answer

The trigonometric identity has been verified. The left side of the identity, \(\tan \left(\theta+\frac{\pi}{4}\right)\), is indeed equal to the right side, \(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\).

Step 1. Identify the appropriate identity to use

The left side of the identity uses the tangent of the sum of two angles. The identity for the tangent of the sum of two angles is given as: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Identify \(a = \theta\) and \(b = \frac{\pi}{4}\). The respective tangent values are \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\tan \frac{\pi}{4} = 1\).

Step 2. Apply the identity

Let's apply the chosen identity:\[\tan\left(\theta+\frac{\pi}{4}\right)=\frac{\frac{\sin \theta}{\cos \theta} + 1}{1 - \frac{\sin \theta}{\cos \theta}}\]

Step 3. Simplify the equation

By making the denominators the same, we can simplify the equation to: \[\tan\left(\theta+\frac{\pi}{4}\right)=\frac{\sin \theta + \cos \theta}{\cos \theta - \sin \theta}\]

Step 4. Conclusion

Thus, we have shown \(\tan \left(\theta+\frac{\pi}{4}\right) = \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\), hence the identity is verified.

Key Concepts

Tangent Sum Identity

Understanding the tangent sum identity is crucial when dealing with trigonometric problems involving the addition of angles. It is expressed as:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
This identity is particularly useful because it allows us to find the tangent of the sum of two angles by knowing the tangent values of the individual angles.

Real-World Applications

Tangent sum identities are applied in various fields such as physics, engineering, and even in geography when determining land areas or navigating. Understanding this identity allows students to handle complex problems, greatly simplifying trigonometric calculations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. Well-known examples include the Pythagorean identities, reciprocal identities, and angle sum and difference identities. These mathematical tools are essential for simplifying trigonometric expressions and solving equations.

Importance in Mathematics

Trigonometric identities underpin many aspects of geometry, calculus, and are essential to solving problems without direct measurement. They enable the derivation of laws that govern the natural world and the design of various technological solutions.

Angle Addition Formulas

Angle addition formulas are trigonometric tools that determine the sine, cosine, and tangent of the sum or difference between two angles. For instance, the formulas for cosine and sine are as follows:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
These formulas are crucial for transformations and calculations in complex geometric problems and provide the foundation for more advanced trigonometry. They extend possibilities in calculus, such as finding the exact value of trigonometric expressions without a calculator.

Understanding Through Patterns

Recognizing patterns in these formulas not only aids memory but also enhances problem-solving skills, helping students tackle mathematical challenges more intuitively.

Identity Verification in Trigonometry

Identity verification in trigonometry involves proving that two trigonometric expressions are equivalent for all values of the variable. This process entails algebraic manipulation, using known identities and logical reasoning to transform one side of the equation to match the other.

Strategy for Verification

Effective strategies include working with one side of the equation until it resembles the other, converting all expressions to sines and cosines, and using algebraic techniques like factoring and common denominators. Mastery of this skill is beneficial for higher-level mathematics, where proving theorems and solving trigonometric equations are common tasks. Through these exercises, students develop analytical thinking that's applicable to various branches of mathematics and science.