Book cover for Precalculus: A Right Triangle Approach

Precalculus: A Right Triangle Approach

Judith A. Beecher, Marvin L. Bittinger, David J. Ellenbogen, Judith A. Penna

ISBN #9780321783967

5th Edition

5,905 Questions

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25,202 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter covers essential trigonometric identities including the Pythagorean and cofunction identities, as well as powerful tools like double-angle and half-angle formulas. These identities, derived using the unit circle and fundamental definitions, are invaluable for simplifying expressions, solving equations, and transitioning to more advanced topics in trigonometry and calculus. Mastery of these concepts enhances analytical problem-solving skills and provides a foundation for understanding the behavior of trigonometric functions both algebraically and graphically.

Learning Objectives

1

Explain and derive key Pythagorean identities from the unit circle.

2

Simplify and verify complex trigonometric expressions using Pythagorean, sum/difference, and cofunction identities.

3

Derive and apply cofunction, double-angle, and half-angle identities in problem-solving.

4

Utilize technological tools to test and confirm the validity of trigonometric identities.

5

Connect the mastery of trigonometric identities to advanced topics in trigonometry and calculus.

Key Concepts

CONCEPT

DEFINITION

Pythagorean Identities

A set of relationships among trigonometric functions derived from the Pythagorean theorem, typically including sin²θ + cos²θ = 1, and related forms involving tangent and secant or cotangent and cosecant.

Cofunction Identities

Identities that relate trigonometric functions of complementary angles (e.g., sin(θ) = cos(90°−θ) or sin(θ) = cos(π/2 − θ) in radians).

Double-Angle Identities

Formulas that express trigonometric functions of twice an angle (e.g., sin(2θ) = 2 sinθ cosθ, cos(2θ) = cos²θ − sin²θ).

Half-Angle Identities

Formulas used to express trigonometric functions of half an angle in terms of the square roots of algebraic expressions (e.g., sin(θ/2) = ±√((1−cosθ)/2)).

Example Problems

Example 1

Multiply and simplify. Check your result using a graphing calculator. $$ (\sin x-\cos x)(\sin x+\cos x) \sin ^{2} x-\cos ^{2} x $$

Example 2

Multiply and simplify. Check your result using a graphing calculator. $$ \tan x(\cos x-\csc x) \quad \sin x-\sec x $$

Example 3

Multiply and simplify. Check your result using a graphing calculator. $$ \cos y \sin y(\sec y+\csc y) \quad \sin y+\cos y $$

Example 4

Multiply and simplify. Check your result using a graphing calculator. $$ (\sin x+\cos x)(\sec x+\csc x) $$

Example 5

Multiply and simplify. Check your result using a graphing calculator. $$ (\sin \phi-\cos \phi)^{2} \quad 1-2 \sin \phi \cos \phi $$
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Step-by-Step Explanations

QUESTION

How do you derive the identity sin²θ + cos²θ = 1 using the unit circle?

STEP-BY-STEP ANSWER:

Step 1: Start with the equation of the unit circle: x² + y² = 1.
Step 2: Recognize that for any angle θ, the coordinates on the unit circle are given by x = cosθ and y = sinθ.
Step 3: Substitute cosθ for x and sinθ for y into the equation: cos²θ + sin²θ = 1.
Step 4: Conclude that sin²θ + cos²θ = 1 is the Pythagorean identity.
Final Answer: sin²θ + cos²θ = 1.

Pythagorean Identity Derivation

QUESTION

How do you show that sin(θ) equals cos(90°−θ)?

STEP-BY-STEP ANSWER:

Step 1: Understand that cofunctions relate functions of complementary angles; complementary angles sum to 90° (or π/2 radians).
Step 2: Recognize that the sine of an angle in a right triangle is the cosine of its complement.
Step 3: Write the identity: sin(θ) = cos(90°−θ) (or sin(θ) = cos(π/2−θ) in radians).
Step 4: Verify by considering a right triangle where one angle is θ and the other is 90°−θ.
Final Answer: sin(θ) = cos(90°−θ).

Cofunction Identity Application

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Common Mistakes

  • Incorrectly substituting values into the unit circle, leading to errors in deriving identities.
  • Overlooking the importance of the sign (positive or negative) when taking square roots, especially in half-angle formulas.
  • Confusing cofunction relationships by misinterpreting complementary angles.
  • Assuming that all related identities are universally applicable without consideration of the specific angle measures or quadrants.
  • Neglecting to verify identity results using technological tools, which can catch algebraic simplification errors.