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

Trigonometric Identities

Calculus

0. Functions

Trigonometric Identities

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2:00 minutes

Problem 1.3.42

Textbook Question

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

cos (3π/2 + x)

Verified step by step guidance

Recognize that the expression cos(3π/2 + x) involves a trigonometric identity for cosine of a sum, which is cos(a + b) = cos(a)cos(b) - sin(a)sin(b).

Identify the values of a and b in the expression: here, a = 3π/2 and b = x.

Substitute these values into the identity: cos(3π/2 + x) = cos(3π/2)cos(x) - sin(3π/2)sin(x).

Recall the exact trigonometric values: cos(3π/2) = 0 and sin(3π/2) = -1.

Substitute these values back into the expression: cos(3π/2 + x) = 0 * cos(x) - (-1) * sin(x), which simplifies to 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. They are essential for simplifying expressions and solving equations in trigonometry. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas, which help express trigonometric functions in various forms.

Angle Addition Formulas

Angle addition formulas are specific trigonometric identities that express the sine and cosine of the sum or difference of two angles. For example, the cosine of the sum of two angles is given by cos(a + b) = cos(a)cos(b) - sin(a)sin(b). These formulas are crucial for transforming expressions involving sums of angles into simpler forms using basic trigonometric functions.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it provides a geometric interpretation of the sine and cosine functions. The coordinates of points on the unit circle correspond to the values of cosine and sine for various angles, making it easier to evaluate trigonometric functions for any angle.