Factoring and Checking Trigonometric Equations

Develop zero-product method and lost roots, squared ratios and both signs.

Lesson 4 of 42 parts Practice Flashcards

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Zero-Product Method and Lost Roots

Objective: Explain zero-product method and lost roots, apply the rule, and justify relevant restrictions.

When a trig equation factors as F(x)G(x)=0, solve F(x)=0 or G(x)=0 and take the union of roots on the requested domain. Dividing by F(x) would remove possible roots where F(x)=0. Overlapping solution families must be counted only once. Check original expressions if a transformation involved multiplication by a denominator or squaring.

Formulas

$FG=0 ⇒ F=0 or G=0$

Worked examples

Try itHow many distinct solutions does sin(7x)(cos(7x)−1)=0 have on 0≤x<2π?Show answer

Step 1: Set each factor equal to zero. Step 2: sin(7x)=0 has 14 roots in the interval; cos(7x)=1 is a subset of those roots. Step 3: The union therefore contains 14 distinct solutions.

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Zero-Product Method and Lost Roots

Objective: Explain zero-product method and lost roots, apply the rule, and justify relevant restrictions.

Formulas

$FG=0 ⇒ F=0 or G=0$

Worked examples

Try itHow many distinct solutions does sin(7x)(cos(7x)−1)=0 have on 0≤x<2π?Show answer

Step 1: Set each factor equal to zero. Step 2: sin(7x)=0 has 14 roots in the interval; cos(7x)=1 is a subset of those roots. Step 3: The union therefore contains 14 distinct solutions.

Squared Ratios and Both Signs

Objective: Explain squared ratios and both signs, apply the rule, and justify relevant restrictions.

If sin²x=q with 0<q<1, solve sin x=√q and sin x=−√q. Each sign generally gives two roots in one complete cycle. Retain the interval and any original domain restrictions. Squaring a prior equation may introduce extraneous candidates, so the final roots must satisfy the starting equation. A nonnegative square does not imply a nonnegative sine.

Formulas

$sin²x=q ⇒ sin x=±√q$

Worked examples

Try itHow many solutions does sin²(7x)=1/4 have on 0≤x<2π?Show answer

Step 1: Take both signs: sin({c}x)=1/2 or −1/2. Step 2: There are four roots per full sine cycle. Step 3: There are 7 cycles, so 28 solutions.

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