TR-32: Intro to Identities and Proofs (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial will guide you through the concepts of identities and proofs in trigonometry, as introduced by Dennis F. Davis. Understanding the difference between equations and identities, along with learning how to prove trigonometric identities, is essential for mastering trigonometry. This tutorial will provide you with a structured approach to tackle these concepts effectively.

Step 1: Understand the Difference Between Equations and Identities

• Equations are statements that assert the equality of two expressions, which can be true for specific values of the variables involved.
• Identities are equations that hold true for all values of the variables within their domain. They are universally valid statements.

Practical Tip

• A common way to identify whether you are dealing with an identity or an equation is to try substituting various values for the variables. If the statement holds for all cases, it is likely an identity.

Step 2: Review Reciprocal Identities

Reciprocal identities are foundational in trigonometric proofs. These identities express trigonometric functions in terms of one another:

• sin(θ) = 1/csc(θ)
• cos(θ) = 1/sec(θ)
• tan(θ) = 1/cot(θ)
• csc(θ) = 1/sin(θ)
• sec(θ) = 1/cos(θ)
• cot(θ) = 1/tan(θ)

Practical Tip

• Memorizing these identities can simplify many proofs, making it easier to manipulate and transform expressions.

Step 3: Learn the Two-Column Proof Format

Formal proofs often utilize a two-column format, which organizes statements and reasons clearly:

1. Statement Column: Write the steps in the proof.
2. Reason Column: Provide justification for each step, citing identities or properties used.

Example Structure

| Statement | Reason | |-----------------------|-----------------------------------------| | Start with sin^2(θ) + cos^2(θ) = 1 | Pythagorean identity | | Replace sin(θ) with 1/csc(θ) | Reciprocal identity | | Continue simplifying | Algebraic manipulation |

Step 4: Practice Simple Proofs Using Reciprocal Identities

To solidify your understanding, try proving the following identity:

• Prove that sin(θ)/cos(θ) = tan(θ).

Steps to Prove

1. Start with the left side: sin(θ)/cos(θ).
2. Apply the reciprocal identity: sin(θ) = 1/csc(θ).
3. Rewrite: (1/csc(θ))/cos(θ).
4. Simplify to show that this equals tan(θ).

Step 5: Complete Extra Problems and Drills

Engage with additional exercises to reinforce your learning. Check out the Extra Problems and Drills video linked in the description for more practice.

Conclusion

This tutorial covered the essential concepts of identities and proofs in trigonometry, focusing on the difference between equations and identities, reciprocal identities, the two-column proof format, and practical proof exercises. To further enhance your skills, practice with various identities and tackle more complex proofs. Consider exploring the additional resources provided in the video description for deeper learning.