Find all solutions on the interval [0,2 π). sin^2 t=cost |...

Name: Problem 32, Chapter 7: Precalculus: An Investigation of Functions
Uploaded: 2021-08-10T22:35:08-08:00
Duration: 2 min 45 s
Channel: Luke Waitrovich
Description: Find all solutions on the interval [0,2 π). sin^2 t=cost

Key Concepts

Quadratic Equations in Trigonometric Functions

Trigonometric equations can often be rewritten in a quadratic form by expressing them in terms of a single function, which allows one to apply algebraic techniques like factoring or using the quadratic formula. This method converts the problem of solving trigonometric equations into solving standard quadratic equations.

Solving Trigonometric Equations on a Specified Interval

Finding solutions within a given interval, such as [0, 2?), involves identifying all angles that satisfy the equation while taking into account the periodicity of trigonometric functions. This step ensures that the full set of valid solutions is captured within the stated domain.

Trigonometric Identities

These are fundamental relationships between trigonometric functions, such as the Pythagorean identity sin²(t) + cos²(t) = 1, which are used to rewrite or simplify expressions in trigonometric equations. They provide a connection between different trigonometric functions and enable the transformation of equations into more workable forms.

Transcript

00:01 Hello and welcome to this video where we're going to solve this equation.

00:04 Sign squared t is equal to cosine t.

00:06 We're going to find all the solutions from 0 to 2 pi.

00:10 Now what we're going to do is we're also going to use a graphing calculator to help us out with this a little bit.

00:14 Now if we wanted to we could actually just type in sine square t and cosine t and we can find their intersections on the graphing calculator.

00:22 But let's at least talk about some of the math that's going on behind here to kind of figure out why we're doing what we're doing.

00:28 So notice that we have two different trig functions.

00:30 Sine squared t and cosine t.

00:32 Well, i'm going to use a basic trigger identity.

00:36 Sign squared plus cosine squared is equal to 1 to kind of help me out with this.

00:42 So if sine squared plus cosine squared is 1, that means that sine squared is equal to 1 minus cosine squared.

00:51 So i'm going to go 1 minus cosine squared t is equal to cosine of t.

01:00 Now, the next step is to get all this on the same side of the equation.

01:04 So i'm going to have 0 equals cosine squared t plus cosine of t minus 1.

01:15 Now, the reason why i do it like this is i want to make sure that this guy was positive.

01:20 Now, here's what we have.

01:21 We have a hidden and embedded quadratic function right here.

01:27 And to make life a little bit easier because like this thing won't factor nicely, let's just go to a graph and help us solve it.

01:35 So i'm going to go over here.

01:37 I'm going to go to my functions...

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