In Exercises 53–62, solve each equation on the interval [0, 2𝝅). tan² x cos x = tan² x

Guida verificata passo dopo passo

Start by writing down the given equation: \(\tan^{2} x \cos x = \tan^{2} x\).

Bring all terms to one side to set the equation equal to zero: \(\tan^{2} x \cos x - \tan^{2} x = 0\).

Factor out the common term \(\tan^{2} x\): \(\tan^{2} x (\cos x - 1) = 0\).

Set each factor equal to zero to find possible solutions: \(\tan^{2} x = 0\) or \(\cos x - 1 = 0\).

Solve each equation separately on the interval \([0, 2\pi)\): - For \(\tan^{2} x = 0\), find \(x\) such that \(\tan x = 0\). - For \(\cos x - 1 = 0\), find \(x\) such that \(\cos x = 1\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Trigonometric Equations

Trigonometric equations involve expressions with trigonometric functions like sine, cosine, and tangent. Solving these equations means finding all angle values within a given interval that satisfy the equation. Understanding how to manipulate and simplify these equations is essential for finding correct solutions.

Video consigliato:

Percorso guidato

4:34

How to Solve Linear Trigonometric Equations

Properties of Tangent and Cosine Functions

The tangent function, tan x, is defined as sin x / cos x and has vertical asymptotes where cos x = 0. Cosine, cos x, oscillates between -1 and 1. Recognizing where these functions are zero or undefined helps in solving equations involving tan² x and cos x, especially when factoring or dividing terms.

Video consigliato:

Percorso guidato

5:08

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Solving Equations on a Specific Interval [0, 2π)

When solving trigonometric equations on the interval [0, 2π), solutions must be found within one full rotation of the unit circle. This requires identifying all angles in this range that satisfy the equation, considering periodicity and the behavior of the trigonometric functions involved.

Video consigliato:

Percorso guidato