Solving Trigonometric Equations Using Algebraic Methods

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Solving Trigonometric Equations using Algebraic Methods

Study Guide

Key Definition

A trigonometric equation is an equation that involves trigonometric functions like $\sin(x)$, $\cos(x)$, and $\tan(x)$.

Important Notes

  • Trigonometric equations can be solved using algebraic methods.
  • Use the zero product property when equations are factorable.
  • For equations in quadratic form, factor if possible or apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Remember that the solutions for $\sin(x)$ are periodic with a period of $2\pi$.
  • Verify solutions by substituting back into the original equation.

Mathematical Notation

$\sin(x)$ is the sine function$\cos(x)$ is the cosine function$\tan(x)$ is the tangent function$\pi$ represents pi, approximately 3.14159$\frac{a}{b}$ represents a fractionRemember to use proper notation when solving problems

Why It Works

By isolating the trigonometric function and using known values of trigonometric functions, we can solve for the variable efficiently.

Remember

Always check the domain and range of the trigonometric functions involved.

Quick Reference

Pythagorean Identity:$\sin^2(x) + \cos^2(x) = 1$Zero Product Property:If $ab = 0$, then $a = 0$ or $b = 0$

Understanding Solving Trigonometric Equations using Algebraic Methods

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Beginner Explanation

When solving trigonometric equations using algebraic methods, we isolate the trig function (like sin(x) or cos(x)) and solve for x using known benchmark values and periodicity.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Solve for $x$ in [0, 2\pi): 2 \sin(x) - 1 = 0 and find the smallest positive solution.

A$x = \frac{\pi}{6}$B$x = \frac{\pi}{4}$C$x = \frac{\pi}{3}$D$x = \frac{\pi}{2}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

A Ferris wheel has a diameter of 50 meters and rotates at a constant speed. Determine the height of a seat above the ground at any angle $\theta$.

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Thinking Challenge

Thinking ExerciseIntermediate

Think About This

If $\tan^2(2x) - 1 = 0$, solve for $x$.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Solve $\cos(2x) = \frac{1}{2}$ for $x$.

A$x = \pm\frac{\pi}{6} + n\pi$B$x = \frac{\pi}{6} + n\pi$C$x = -\frac{\pi}{6} + n\pi$D$x = \frac{\pi}{3} + n\pi$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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