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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2:15 minutes

Problem 1.3.40

Textbook Question

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

sin (2π − x)

Verified step by step guidance

Recognize that the expression involves the sine of a difference: sin(2π - x). This can be simplified using the sine subtraction formula: sin(a - b) = sin(a)cos(b) - cos(a)sin(b).

Substitute a = 2π and b = x into the formula: sin(2π - x) = sin(2π)cos(x) - cos(2π)sin(x).

Recall the values of sin(2π) and cos(2π). Since 2π is a full rotation in the unit circle, sin(2π) = 0 and cos(2π) = 1.

Substitute these values into the expression: sin(2π - x) = 0 * cos(x) - 1 * sin(x).

Simplify the expression: sin(2π - x) = -sin(x).

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is the sine and cosine of complementary angles, which states that sin(π - x) = sin(x) and cos(π - x) = -cos(x). These identities are essential for simplifying expressions involving trigonometric functions.

Sine Function Properties

The sine function, denoted as sin(x), is a periodic function that represents the y-coordinate of a point on the unit circle corresponding to an angle x. It has specific properties, such as sin(2π - x) = sin(x), which reflects the symmetry of the sine function about the y-axis. Understanding these properties helps in transforming trigonometric expressions.

Angle Subtraction

Angle subtraction refers to the process of finding the sine or cosine of an angle expressed as the difference between two angles. In this case, sin(2π - x) can be analyzed using the periodic nature of the sine function, which allows us to express it in terms of sin(x) and cos(x). Recognizing how angles relate to one another is crucial for simplifying trigonometric expressions.

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Textbook Question

[Technology Exercise]

You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how long should the arc be?

Textbook Question

Radians and Degrees

On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?

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

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

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.

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