For the following exercises, find all exact solutions on the interval [0,2 π) 2 sin(2 x)+√(2)=0

Name: Problem 424, Chapter 7: Precalculus
Uploaded: 2020-06-28T22:57:24-08:00
Duration: 1 min 11 s
Channel: Aamir Mithaiwala
Description: For the following exercises, find all exact solutions on the interval [0,2 π) 2 sin(2 x)+√(2)=0

step 1 So for this problem, let's substitute, let's say, let's say you is equal to 2x. And so what we'r

For the following exercises, find all exact solutions on the...

Key Concepts

Trigonometric Equations

Trigonometric equations involve functions such as sine, cosine, and tangent, and require finding all angle solutions that satisfy a condition. These equations are tackled by using identities, transformations, and properties of periodicity, ensuring that all solutions over the relevant interval are identified.

Sine Function

The sine function is a fundamental trigonometric function characterized by its periodicity, range, and specific symmetry properties. Understanding its behavior—including its periodic nature and its positive and negative values in different quadrants—is essential for solving equations that involve sine, as the solutions often rely on identifying reference angles within one period.

General Solution of Sine Equations

To solve a sine equation, one typically begins by isolating the sine term and then determining the reference angle that yields the given sine value. The general solution for a sine equation accounts for the periodicity by including all angles that differ by full cycles, usually expressed in the form of an initial solution plus integer multiples of the period.

Reference Angle

A reference angle is the acute angle that a given angle makes with the horizontal axis. It is used as a basis to find all angles that yield the same sine value, by considering the symmetry of the sine function in the unit circle. The reference angle simplifies the process of identifying the correct angles in the appropriate quadrants.

Interval Restriction

When finding solutions to trigonometric equations, it is often necessary to limit the answers to a specific interval, such as [0, 2?). This involves filtering the general solution set to only include those angles that lie within the prescribed bounds of the interval, ensuring the final answer is exact and applicable.

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