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Solving Trigonometric Equations using Trigonometric Identities

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Solving Trigonometric Equations using Trigonometric Identities

Study Guide

Key Definition

Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variables involved, such as $\sin^2(x) + \cos^2(x) = 1$.

Important Notes

Extraneous solutions can occur when squaring both sides of an equation.
Use identities like $\sin^2(x) = 1 - \cos^2(x)$ to simplify equations.
Check solutions within the given interval.
Convert all terms to a single trigonometric function when possible.
Verify solutions by substituting back into the original equation.

Mathematical Notation

$\sin(x)$ represents the sine function

$\cos(x)$ represents the cosine function

$\tan(x)$ represents the tangent function

$+$ represents addition

$-$ represents subtraction

Angles are in radians by default; when converting to degrees, ensure consistency in units.

Why It Works

Using trigonometric identities simplifies complex equations into solvable forms, such as rewriting $2 \sin^2(x) = 2 + \cos(x)$ using identities to isolate terms.

Remember

Key formulas include $\sin^2(x) + \cos^2(x) = 1$ and $\tan^2(x) + 1 = \sec^2(x)$.

Quick Reference

Pythagorean Identity:$\sin^2(x) + \cos^2(x) = 1$

Angle Sum Identity:$\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)$

Understanding Solving Trigonometric Equations using Trigonometric Identities

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

At the beginner level, solving trigonometric equations starts with recognizing and applying basic identities such as $\sin^2(x) + \cos^2(x) = 1$. For example, when you see $\sin^2(x)$ in an equation, you can replace it with $1 - \cos^2(x)$ to simplify and solve for $x$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the value of $\sin^2(x) + \cos^2(x)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a ramp with an angle $\theta$, where $\tan(\theta) = \frac{3}{4}$. Calculate the angle $\theta$ (in degrees, where 0° ≤ θ < 90°).

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove the identity $1 + \tan^2(x) = \sec^2(x)$.

Challenge Quiz

Single Choice Quiz

Advanced

Solve $2\sin^2(x) = 2 + \cos(x)$ for $x$ in the interval $[0, 2\pi)$.

Recap

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Review key concepts and takeaways