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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 fraction

Remember 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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Video explanation of this concept

Beginner

Start here! Easy to understand

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.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

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

Thinking Challenge

Thinking Exercise

Intermediate

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 $\theta$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

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

Challenge Quiz

Single Choice Quiz

Advanced

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

Recap

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