Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 3.5.57

Chapter 3, Problem 3.5.57

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). cot x (tan x - 1) = 0

Verified step by step guidance

Start by analyzing the given equation: \(\cot x (\tan x - 1) = 0\). Since this is a product equal to zero, use the zero product property which states that if \(AB = 0\), then either \(A = 0\) or \(B = 0\).

Set each factor equal to zero separately: 1) \(\cot x = 0\) 2) \(\tan x - 1 = 0\)

Solve the first equation \(\cot x = 0\). Recall that \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot x = 0\) when \(\cos x = 0\) (and \(\sin x \neq 0\)). Find all \(x\) in \([0, 2\pi)\) where \(\cos x = 0\).

Solve the second equation \(\tan x - 1 = 0\) which simplifies to \(\tan x = 1\). Find all \(x\) in \([0, 2\pi)\) where the tangent of \(x\) equals 1.

Combine all solutions from both equations and ensure they lie within the interval \([0, 2\pi)\). These combined values of \(x\) will be the solutions to the original equation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Trigonometric Functions and Their Domains

Understanding the definitions and properties of cotangent and tangent functions is essential. Cotangent is the reciprocal of tangent, and both are periodic with period π. Knowing their domains and where they are undefined helps in solving equations and identifying valid solutions within the interval [0, 2π).

Zero-Product Property

The zero-product property states that if a product of two factors equals zero, then at least one of the factors must be zero. This principle allows us to split the equation cot x (tan x - 1) = 0 into two simpler equations: cot x = 0 and tan x - 1 = 0, which can be solved separately.

Solving Trigonometric Equations on a Specific Interval

When solving trigonometric equations, it is important to find all solutions within the given interval, here [0, 2π). This involves finding general solutions using known values of trig functions and then restricting them to the specified domain, considering periodicity and any domain restrictions.

Recommended video:

4:34

How to Solve Linear Trigonometric Equations

Related Practice

Textbook Question

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 4 cos² x - 1 = 0

views

Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). sin x + 2 sin x cos x = 0

views

Textbook Question

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). (2 cos x + √ 3) (2 sin x + 1) = 0

views

Textbook Question

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 2 cos² x + sin x - 1 = 0

Textbook Question

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. tan(θ/2)

Textbook Question

In Exercises 47–54, use the figures to find the exact value of each trigonometric function. 2sin(θ/2)cos(θ/2)

views