Solve the following equations for x (0 ≤x<2 π) sin3 x=-(1)/(2)

Name: Problem 4, Chapter 32: Calculus With Trigonometry and Analytic Geometry
Uploaded: 2020-06-05T20:17:40-08:00
Duration: 4 min 37 s
Channel: Ryan Williams
Description: Solve the following equations for x (0 ≤x<2 π) sin3 x=-(1)/(2)

Solve the following equations for x (0 ≤x<2 π) sin3...

Key Concepts

Periodicity and Angle Transformation

When dealing with equations like sin(3x) = k, a variable transformation is applied where the argument of sine is multiplied by a constant. This alters the period of the sine function, reducing it to 2? divided by that constant. Understanding how to adjust the solution set according to the modified period is crucial in finding all valid solutions within a specified interval.

Reference Angle

The reference angle is the acute angle that the terminal side of an angle makes with the horizontal axis. When solving trigonometric equations, the reference angle helps in determining all the angles that yield a given sine value by accounting for the symmetry of the sine function in different quadrants.

Sine Function

The sine function is a periodic trigonometric function that oscillates between -1 and 1. Its behavior and symmetry properties, such as being odd and periodic with period 2?, are essential for solving equations where the sine of an angle is set equal to a constant.

General Solution of Sine Equations

To solve equations of the form sin(?) = k, where k is a constant, one uses the general solution formula. This incorporates the idea that sine reaches the same value at two distinct points in each period, and thus all solutions can be expressed by adding integer multiples of the period (2?) to these principal angles.

Transcript

00:01 To solve this equation, it's easier to first think about instead of the sign of 3x, for what angles the sine of theta is equal to the square root of 2 over 2.

00:15 And here, we can find that, from the unit circle, that the sign of theta, when theta is pi over 4, is equal to the square root 2 over 2, and also for when theta is equal to 3 pi over 4.

00:33 So pi over 4, plus if you go around the circle, 2 pi radiance, you'll end up back at the same point.

00:41 So 2 pi times some integer n, or the same thing with 3 pi over 4, plus 2 pi times some integer n.

00:53 These are the values of theta in which this is true.

00:56 Now we just have to relate theta in x.

01:00 Since we see it that the sign of theta, we solve for when that is equal to square to 2 over 2.

01:05 We want to know when the sign of 3x is equal to that.

01:08 So we therefore have to know when 3x is equal to theta, when the angles are equal.

01:14 Or that's just when x is equal to theta over 3.

01:20 If we divide the values that we found up here divided by 3, we get pi over 12 plus 2 pi over 3 times some integer n and pi over 4 plus 2 pi over 3 times some integer n are our solutions for x.

01:46 However, x is bounded between 0 and 2 pi.

01:51 So we're going to have a very limited amount of solutions.

01:53 And to figure out what our solutions are, we just have to go through...

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