Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter introduces trigonometric functions, emphasizing their periodic nature and applications in modeling real-world phenomena. Key topics include the conversion between degree and radian measures, the definitions and usage of sine, cosine, tangent and their reciprocals via the unit circle and right triangles, and the graphical transformations of trigonometric functions. Understanding these concepts is crucial for further studies in calculus, physics, and engineering applications.

Learning Objectives

1

Describe the characteristics of trigonometric functions and how they model periodic phenomena.

2

Convert between degree and radian measures and understand the concept of standard position.

3

Define and compute the six trigonometric functions using right triangles and the unit circle.

4

Graph and transform sine and cosine functions by identifying amplitude, period, phase shift, and vertical shift.

5

Apply trigonometric concepts to real-world problems in physics, biology, and economics.

Key Concepts

CONCEPT

DEFINITION

Periodic Function

A function that repeats its values in regular intervals or periods. For trigonometric functions, sine and cosine have a period of 2π and tangent has a period of π.

Angle in Standard Position

An angle whose vertex is at the origin of a coordinate system and whose initial side lies along the positive x-axis.

Degree Measure

A system for measuring angles where a full rotation is 360°. One degree is 1/360 of a full rotation.

Radian Measure

A measure of an angle obtained by dividing the arc length on a unit circle by the radius. One radian is the angle that subtends an arc equal in length to the radius, with 2Ï€ radians making a full circle.

Trigonometric Functions

Functions defined from the coordinates of a point on the terminal side of an angle in standard position. They include sine (y/r), cosine (x/r), tangent (y/x), and their reciprocals: cosecant (r/y), secant (r/x), and cotangent (x/y).

Special Triangles

Right triangles with specific angle measures (such as 30°–60°–90° and 45°–45°–90°) used to determine exact values of trigonometric functions.

Transformation Parameters

Constants a, b, c, and d in functions like y = a sin(bx - c) + d that define the amplitude (a), period (2Ï€/b), phase shift (c/b), and vertical shift (d) of the trigonometric graph.

Example Problems

Example 1

Convert the following degree measures to radians. Leave answers as multiples of $\pi$. $$ 65^{\circ} $$

Example 2

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$ $$90^{\circ}$$

Example 3

Convert the following degree measures to radians. Leave answers as multiples of $\pi$. $$ 95^{\circ} $$

Example 4

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$ $$135^{\circ}$$

Example 5

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$ $$270^{\circ}$$
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Step-by-Step Explanations

QUESTION

Convert 45° to radians.

STEP-BY-STEP ANSWER:

Step 1: Recall the conversion factor that 180° = π radians.
Step 2: Multiply 45° by π/180.
Step 3: Simplify: 45π/180 = π/4.
Final Answer: 45° = π/4 radians.

Degree to Radian Conversion

QUESTION

For a point P(8, 15) on the terminal side of an angle with r = 17, calculate sin θ, cos θ, and tan θ.

STEP-BY-STEP ANSWER:

Step 1: Calculate r using the Pythagorean theorem if needed: r = √(8² + 15²) = 17.
Step 2: sin θ = y/r = 15/17.
Step 3: cos θ = x/r = 8/17.
Step 4: tan θ = y/x = 15/8.
Final Answer: sin θ = 15/17, cos θ = 8/17, tan θ = 15/8.

Finding Trigonometric Function Values from a Point

QUESTION

Graph y = 3 sin(2x - π) + 1 and identify the amplitude, period, phase shift, and vertical shift.

STEP-BY-STEP ANSWER:

Step 1: Amplitude is |a| = 3.
Step 2: Period is computed as 2π divided by |b|, so period = 2π/2 = π.
Step 3: Phase shift is c/b = (π)/2 (right shift if written in standard form y = 3 sin(2(x - π/2)) + 1).
Step 4: Vertical shift is d = 1.
Final Answer: Amplitude = 3, Period = π, Phase shift = π/2 to the right, Vertical shift = 1 upward.

Graphing a Transformed Sine Function

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Common Mistakes

  • Failing to switch the calculator mode between degrees and radians, leading to erroneous results.
  • Misidentifying the quadrant of an angle in standard position, resulting in sign errors for trigonometric functions.
  • Incorrectly converting between degrees and radians by forgetting the conversion factor ?/180.
  • Overlooking the periodic properties of trigonometric functions and not recognizing equivalent angles differing by multiples of 2?.
  • Confusing the roles of amplitude, period, phase shift, and vertical shift in graph transformations.