/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Show that \(\cos ^{2}(z)+\sin ^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 4

Show that \(\cos ^{2}(z)+\sin ^{2}(z)=1\)

Short Answer

Expert verified
Using the Pythagorean identity \(\sin^2(z) + \cos^2(z) = 1\), we see that the given expression \(\cos^{2}(z) + \sin^{2}(z)\) satisfies the identity. Therefore, it is proven that \(\cos^{2}(z) + \sin^{2}(z) = 1\).

Step by step solution

01

Pythagorean identity

The Pythagorean identity states that for any angle \(z\), we have \(\sin^2(z) + \cos^2(z) = 1\). We are given the expression \(\cos ^{2}(z)+\sin ^{2}(z)\) and want to show that it is equal to \(1\). Since we have already stated the Pythagorean identity, we directly observe that \(\cos ^{2}(z)+\sin ^{2}(z) = 1\). Thus, the given expression is equal to \(1\), and the identity is proved.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions are fundamental in mathematics, especially in the study of triangles and modeling periodic phenomena. They describe the relationship between the angles and sides of a right triangle. The main trigonometric functions include:
  • Sine (\( \sin \) ): Ratio of the opposite side to the hypotenuse.
  • Cosine (\( \cos \) ): Ratio of the adjacent side to the hypotenuse.
  • Tangent (\( \tan \) ): Ratio of the sine to the cosine, or the opposite side over the adjacent side.
These functions help describe angles in terms of quantities which are easier to work with in mathematical calculations and are extensible beyond just right triangles using the unit circle, which fully conveys their periodic nature. Understanding these functions opens the door to exploring more complex properties and equations such as trigonometric identities.
Trigonometric Identities
Trigonometric identities are equations containing trigonometric functions that hold true for all values of the variable where they are defined. They allow for the simplification of complex trigonometric expressions and the solving of trigonometric equations.Some essential types include:
  • Pythagorean Identities: Involves the square of sine and cosine, a key identity being \(\sin^2(z) + \cos^2(z) = 1\).
  • Angle Sum and Difference Identities: These involve \(\sin(A \pm B)\) and \(\cos(A \pm B)\).
  • Double Angle Identities: Consider terms like \(\sin(2A)\) and \(\cos(2A)\).
Among these, the Pythagorean identity uniquely associates sine and cosine in a way rooted deeply in geometry, as it derives from the geometric Pythagorean theorem directly applied to the unit circle.
Cosine and Sine
Cosine and sine are the backbone of trigonometric functions, pivotal in describing circular and oscillatory motion. Sine Function (\( \sin \) ):
- Measures the vertical component in the unit circle.- Starts at 0 and peaks at 1.
Cosine Function (\( \cos \) ):
- Measures the horizontal component in the unit circle.- Starts at 1 and decreases to 0.
The unique quality of both functions is their periodic nature, repeating every \(2\pi\) radians, which means they are excellent for modeling periodic processes like sound waves, light waves, and tides. Additionally, when squared and added (\( \cos^2(z) + \sin^2(z) = 1 \)), they exemplify the Pythagorean identity, reinforcing their fundamental relationship and importance.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Verify that \(\cos (z)\) and \(\sin (z)\) satisfy the Cauchy-Riemann equations and use the proposition to evaluate their derivatives.

With \(z^{w} \equiv e^{w \ln (z)}\) for complex \(z\) and \(w\) compute \(\sqrt{i}\)

Express \(\left|e^{z}\right|\) in terms of \(x\) and/or \(y\).

Find the real and imaginary parts of \(\cos (z)\) and \(\sin (z)\)

Confirm that \(e^{\ln (z)}=z\) and \(\ln \left(e^{z}\right)=z\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.