/*! 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 28 Verify the identity. $$ \tan... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 28

Verify the identity. $$ \tan \left(\theta+\frac{\pi}{2}\right)=-\cot (\theta) $$

Short Answer

Expert verified
The identity \(\tan(\theta+\frac{\pi}{2})=-\cot(\theta)\) is verified using properties of angle shifts in trigonometry.

Step by step solution

01

Recall the Tangent Addition Formula

The formula for tangent of a sum is \[\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\]. Here, A is \(\theta\) and B is \(\frac{\pi}{2}\).
02

Substitute into the Formula

Substitute \(A = \theta\) and \(B = \frac{\pi}{2}\) into the tangent addition formula:\[\tan\left(\theta + \frac{\pi}{2}\right) = \frac{\tan\theta + \tan\frac{\pi}{2}}{1 - \tan\theta \tan\frac{\pi}{2}}\].
03

Analyze \(\tan\frac{\pi}{2}\)

Recall that \(\tan\frac{\pi}{2}\) is undefined because cosine at this angle is zero, causing division by zero in the definition of tangent.
04

Use an Alternative Identity

Use the identity \(\tan\left(\theta + \frac{\pi}{2}\right) = -\cot\theta\) directly based on complementary angle properties of tangent and cotangent. Because adding \(\frac{\pi}{2}\) to a function's angle shifts it orthogonally on the unit circle, this results in negatives of the reciprocal functions.
05

Conclude the Identity Verification

Therefore, the identity \(\tan\left(\theta + \frac{\pi}{2}\right) = -\cot(\theta)\) is verified by understanding how tangent and cotangent relate as complementary functions. Since the expression simplifies to \(-\cot(\theta)\), the identity is confirmed.

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.

Tangent Addition Formula
The tangent addition formula is a helpful tool in trigonometry, especially when dealing with sums of angles. It provides a way to calculate the tangent of the sum of two angles. The formula is:
  • \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
This formula works by expressing the tangent of a combined angle in terms of the tangents of the individual angles. When you plug in specific values for \(A\) and \(B\), like in this problem with \( \theta \) and \( \frac{\pi}{2} \), you can manipulate the formula to solve or verify identities.
Cotangent
The cotangent of an angle is closely related to the tangent but is essentially its reciprocal. If \( \tan(\theta) = \frac{\sin (\theta)}{\cos (\theta)} \), then the cotangent is:
  • \( \cot(\theta) = \frac{\cos (\theta)}{\sin (\theta)} \)
This means that whenever tangent is defined, cotangent is as well, except where sine is zero, resulting in an undefined expression. Understanding where these functions are undefined is key, especially when verifying identities like the one at hand.
Complementary Angles
Complementary angles always sum up to \( \frac{\pi}{2} \) radians or 90 degrees. In trigonometry, functions like sine and cosine have known relationships with angles that complement to \( \frac{\pi}{2} \). These relationships extend to a complementary identity in tangent:
  • When dealing with \( \tan(\theta + \frac{\pi}{2}) \), the identity simplifies to \( -\cot(\theta) \) because of the perpendicular shift on the unit circle.
The complementary angle relationships are crucial for understanding how angle manipulations affect trigonometric functions.
Unit Circle
The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1, centered at the origin of the coordinate plane. Every point on the circle can be associated with an angle, where:
  • The x-coordinate represents the cosine of that angle.
  • The y-coordinate represents the sine of the angle.
For tangent and cotangent, these are derived from sine and cosine:
  • \( \tan(\theta) = \frac{y}{x} \)
  • \( \cot(\theta) = \frac{x}{y} \)
Using the unit circle helps visualize why adding \( \frac{\pi}{2} \) to an angle results in transformations, leading us to understand identities like \( \tan(\theta + \frac{\pi}{2}) = -\cot(\theta) \). It's an invaluable tool for making sense of trigonometric relationships.

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

Find the domain of the given function. Write your answers in interval notation. $$ f(x)=\arctan (\ln (2 x-1)) $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}\left(\sec \left(\frac{11 \pi}{12}\right)\right)\)

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \cos \left(\arcsin \left(\frac{x}{2}\right)\right) $$

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \cos (2 \arcsin (4 x)) $$

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \sec (\arctan (2 x)) \tan (\arctan (2 x)) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

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.