Solve each trig equation below for [tex]$0 \leq x\ \textless \ \pi$[/tex].
[tex]$\tan x=\tan x \sin ^2 x$[/tex]
Answer (2)
Rewrite the equation: tan x − tan x sin 2 x = 0 .
Factor out tan x : tan x ( 1 − sin 2 x ) = 0 .
Use the identity 1 − sin 2 x = cos 2 x : tan x cos 2 x = 0 .
Simplify to sin x cos x = 0 , so sin x = 0 or cos x = 0 . The solution is 0 .
Explanation
Understanding the Problem We are asked to solve the trigonometric equation tan x = tan x sin 2 x for 0 ≤ x < π . This means we need to find all values of x within the interval [ 0 , π ) that satisfy the given equation.
Rewriting the Equation First, let's rewrite the equation by subtracting tan x sin 2 x from both sides: tan x − tan x sin 2 x = 0 Now, we can factor out tan x from the left side: tan x ( 1 − sin 2 x ) = 0
Using Trigonometric Identities We know the trigonometric identity 1 − sin 2 x = cos 2 x . Substituting this into the equation, we get: tan x cos 2 x = 0 Now, we can express tan x as c o s x s i n x : cos x sin x cos 2 x = 0 Simplifying, we have: sin x cos x = 0
Solving for \sin x = 0 and \cos x = 0 This equation is satisfied if either sin x = 0 or cos x = 0 . Let's consider each case separately.
Case 1: Solving sin(x) = 0 Case 1: sin x = 0 . In the interval [ 0 , π ) , the solution is x = 0 . (Note that x = π is not included because the interval is [ 0 , π ) .)
Case 2: Solving cos(x) = 0 Case 2: cos x = 0 . In the interval [ 0 , π ) , the solution is x = 2 π .
Checking for Extraneous Solutions However, we must check for extraneous solutions. The original equation is tan x = tan x sin 2 x . Since tan x = c o s x s i n x , tan x is undefined when cos x = 0 . Therefore, x = 2 π is not a valid solution because tan ( 2 π ) is undefined.
Final Solution Therefore, the only solution to the equation tan x = tan x sin 2 x in the interval [ 0 , π ) is x = 0 .
Examples
Trigonometric equations are used in physics to model oscillations and waves. For example, the motion of a pendulum or the propagation of light can be described using trigonometric functions. Solving trigonometric equations allows us to determine specific points in time or space where these phenomena exhibit certain properties, such as maximum displacement or zero amplitude. This is crucial in fields like engineering and signal processing, where understanding and manipulating wave behavior is essential.
The solution to the equation tan x = tan x sin 2 x for 0 ≤ x < π is x = 0 only, as x = 2 π is extraneous. Therefore, the final answer is x = 0 .
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