Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Chapter 5 introduces the fundamental trigonometric identities and demonstrates how to leverage these identities to evaluate functions, simplify expressions, factor trigonometric terms, and solve equations. Mastery of these skills is crucial for advanced studies in trigonometry and calculus, and these methods are widely applicable in solving real-world problems, from physics to engineering.

Learning Objectives

1

Identify and state the fundamental trigonometric identities including reciprocal, quotient, Pythagorean, cofunction, and even/odd identities.

2

Use these identities to evaluate unknown trigonometric functions given one or more function values and quadrant information.

3

Simplify and rewrite complex trigonometric expressions by factoring, finding a common denominator, and substituting with sine and cosine.

4

Develop additional trigonometric identities and verify identities methodically.

5

Apply these techniques to solve trigonometric equations and model real?world problems such as friction and rate of change.

Key Concepts

CONCEPT

DEFINITION

Fundamental Trigonometric Identities

Basic relationships between sine, cosine, and other trigonometric functions, including reciprocal, quotient, Pythagorean, cofunction, and even/odd identities.

Reciprocal Identities

Identities that relate a trigonometric function to its reciprocal (e.g., sin u and csc u, cos u and sec u, tan u and cot u).

Pythagorean Identities

Identities derived from the Pythagorean theorem, such as sin^2 u + cos^2 u = 1, 1 + tan^2 u = sec^2 u, and 1 + cot^2 u = csc^2 u.

Quotient Identities

Identities that express tangent and cotangent as ratios of sine to cosine and cosine to sine respectively.

Cofunction Identities

Identities linking trigonometric functions of complementary angles, such as sin(90° − u) = cos u.

Even/Odd Identities

Identities that show which trigonometric functions are even (e.g., cos u, sec u) and which are odd (e.g., sin u, tan u, csc u, cot u), affecting the sign when the angle is negated.

Example Problems

Example 1

Fill in the blank to complete the trigonometric identity. $ \dfrac{\sin u}{\cos u} $= ________

Example 2

Fill in the blank to complete the trigonometric identity. $ \dfrac{1}{\csc u} $= ________

Example 3

Fill in the blank to complete the trigonometric identity. $ \dfrac{1}{\tan u} $= ________

Example 4

Fill in the blank to complete the trigonometric identity. $ \dfrac{1}{\cos u} $= ________

Example 5

Fill in the blank to complete the trigonometric identity. $ 1 + $ ________ =$ \csc^2 u $
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Step-by-Step Explanations

QUESTION

Given that sin u = -4/5 and u is in Quadrant III, find cos u, tan u, csc u, sec u, and cot u.

STEP-BY-STEP ANSWER:

Step 1: Use the Pythagorean identity sin^2 u + cos^2 u = 1. Substitute sin u = -4/5: (-4/5)^2 + cos^2 u = 1, which gives 16/25 + cos^2 u = 1.
Step 2: Solve for cos^2 u: cos^2 u = 1 - 16/25 = 9/25.
Step 3: Determine cos u. Since u is in Quadrant III where cosine is negative, cos u = -3/5.
Step 4: Compute tan u = sin u/cos u = (-4/5)/(-3/5) = 4/3.
Step 5: Find the reciprocals: csc u = 1/sin u = -5/4, sec u = 1/cos u = -5/3, and cot u = 1/tan u = 3/4.
Final Answer: sin u = -4/5, cos u = -3/5, tan u = 4/3, csc u = -5/4, sec u = -5/3, and cot u = 3/4.

Evaluating Trigonometric Functions

QUESTION

Simplify the expression sin x cos^2 x - sin x.

STEP-BY-STEP ANSWER:

Step 1: Factor out the common monomial factor sin x: sin x (cos^2 x - 1).
Step 2: Recognize that cos^2 x - 1 is equivalent to - sin^2 x (using the identity sin^2 x + cos^2 x = 1).
Step 3: Substitute to obtain: sin x * (- sin^2 x) = - sin^3 x.
Final Answer: The expression simplifies to - sin^3 x.

Simplifying a Trigonometric Expression

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Common Mistakes

  • Incorrectly choosing the sign when taking square roots from Pythagorean identities, leading to wrong function values.
  • Overlooking domain restrictions and quadrant information which affect the sign and value of trigonometric functions.
  • Failing to factor common terms before applying identities, which can complicate or obscure the path to simplification.
  • Mixing up reciprocal and quotient identities, resulting in misapplied formulas.