Solving Trigonometric Equations

Develop roots on a specified interval, general solution families.

Lesson 3 of 42 parts Practice Flashcards

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Roots on a Specified Interval

Objective: Explain roots on a specified interval, apply the rule, and justify relevant restrictions.

A periodic equation can have multiple roots within a stated interval and infinitely many over the real line. Find base-angle solutions, include quadrant alternatives, then account for the input multiplier and full-period shifts. For cos(Bx)=0, B>0, use Bx=π/2+kπ. Check interval endpoints explicitly; [0,2π) includes zero and excludes 2π.

Formulas

$cos(Bx)=0 ⇒ Bx=π/2+kπ$

Worked examples

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

Step 1: As x runs through the interval, 7x runs through 7 full turns. Step 2: Cosine is zero twice per full turn. Step 3: The total is 2·7=14.

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Roots on a Specified Interval

Objective: Explain roots on a specified interval, apply the rule, and justify relevant restrictions.

Formulas

$cos(Bx)=0 ⇒ Bx=π/2+kπ$

Worked examples

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

Step 1: As x runs through the interval, 7x runs through 7 full turns. Step 2: Cosine is zero twice per full turn. Step 3: The total is 2·7=14.

General Solution Families

Objective: Explain general solution families, apply the rule, and justify relevant restrictions.

Sine vanishes at integer multiples of π, so sin x=0 has solution x=kπ for k∈ℤ. General solutions to sin x=a use both principal-angle and supplementary-angle families, plus 2πk shifts where appropriate. A family compactly records infinitely many solutions; an interval answer is obtained by restricting the integer index. Keep radians and degrees consistent.

Formulas

$sin x=0 ⇔ x=kπ, k∈ℤ$

Worked examples

Try itThe nonnegative solutions of sin x=0 are x=nπ. What is the 8st solution, counting 0 as the first? Give the coefficient of π.Show answer

Step 1: All solutions are integer multiples of π. Step 2: The first is n=0; the 8st uses n=7. Step 3: The requested solution is 7π.

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