Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Introduction to Trigonometric Functions

Calculus

0. Functions

Introduction to Trigonometric Functions

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1:45 minutes

Problem 56a

Textbook Question

Assume that a particle’s position on the x-axis is given by

x = 3 cos t + 4 sin t,

where x is measured in feet and t is measured in seconds.

a. Find the particle’s position when t = 0, t = π/2, and t = π.

Verified step by step guidance

To find the particle's position at a specific time, substitute the given values of t into the position function x = 3 cos t + 4 sin t.

For t = 0, substitute t = 0 into the equation: x = 3 cos(0) + 4 sin(0). Recall that cos(0) = 1 and sin(0) = 0.

For t = π/2, substitute t = π/2 into the equation: x = 3 cos(π/2) + 4 sin(π/2). Recall that cos(π/2) = 0 and sin(π/2) = 1.

For t = π, substitute t = π into the equation: x = 3 cos(π) + 4 sin(π). Recall that cos(π) = -1 and sin(π) = 0.

Evaluate each expression to find the particle's position at t = 0, t = π/2, and t = π using the trigonometric values provided.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in describing periodic phenomena. In this context, they represent the particle's oscillatory motion along the x-axis. Understanding how to evaluate these functions at specific angles, like 0, π/2, and π, is crucial for determining the particle's position at given times.

Evaluating Trigonometric Expressions

To find the particle's position at specific times, we need to evaluate the trigonometric expression x = 3 cos t + 4 sin t. This involves substituting the given values of t into the expression and calculating the result. Familiarity with the unit circle and the values of sine and cosine at key angles is essential for this process.

Position Function

The position function x = 3 cos t + 4 sin t describes the particle's location on the x-axis over time. It combines the effects of two harmonic motions, each with its amplitude and phase. Understanding how to interpret and manipulate this function allows us to predict the particle's position at any given time t.

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Related Practice

Textbook Question

Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.

Textbook Question

Refer to the given figure. Write the radius r of the circle in terms of α and θ.

Textbook Question

In Exercises 39–42, express the given quantity in terms of sin x and cos x.

sin (2π − x)

Textbook Question

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Find the spring’s displacement when t = 0, t = π/3, and t = 3π/4.

Textbook Question

Computer Explorations

Use a CAS to perform the following steps in Exercises 55–62.

a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.

xy³ + tan(x + y) = 1, P(π/4, 0)

Multiple Choice

Convert the angle $540°$ from degrees to radians.