In 3-10, find the exact values of θin the interval 0^∘ ≤θ≤360^∘ that make each equation true. 3

In 3-10, find the exact values of θin the interval 0^∘...

Key Concepts

Quadratic Equations in Trigonometric Functions

Some trigonometric equations can be rearranged into quadratic form in a trigonometric function, such as cos? or sin?. This permits the use of algebraic techniques like factoring or the quadratic formula to find potential solutions.

Unit Circle and Special Angles

The unit circle encompasses information about the exact values of trigonometric functions at commonly encountered angles. Understanding these values is essential for determining the exact solutions of trigonometric equations within specific intervals.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. These identities allow the conversion between different forms of expressions, making it easier to simplify and solve complex trigonometric equations.

Double-Angle Formula

The double-angle formulas express trigonometric functions of double angles, such as cos(2?), in terms of the trigonometric functions of the single angle ?. This is useful for rewriting and simplifying equations that involve multiples of the variable.

Solving Trigonometric Equations

Solving trigonometric equations involves a combination of applying trigonometric identities, converting equations into more manageable forms like quadratic equations, and then using techniques from algebra along with knowledge of the periodicity and symmetry of trigonometric functions to obtain all solutions within a given domain.

Transcript

00:01 All right, so we're going to find the exact values for theta between zero and 360.

00:05 So this one's a little bit of a challenge.

00:08 First, i always look at these sort of odd with theta not by itself, and we're going to replace this with a double angle identity.

00:17 So you've got to keep that three, and a cosine 2 theta is also equal to cosine squared theta minus a sine squared theta.

00:28 And then we have minus 4 cosine squared theta plus 2 equal to 0.

00:36 And so now we have to distribute this out.

00:39 So we have a 3 cosine squared theta minus 3 sine squared theta minus a 4 cosine squared theta plus 2, oops, plus 2 equal to 0.

00:54 And now let's see, we're going to combine this cosine here and this cosine.

00:58 Here.

00:59 So i have a minus four and a plus three.

01:02 So that means i have a minus cosine squared theta and a minus three sine squared theta plus two equal to zero.

01:13 The first thing i'm going to do here is multiply this whole thing.

01:16 And you do both sides, but when you multiply it by zero, it stays zero.

01:19 I'm going to multiply this by a negative one.

01:22 So what i get is cosine squared theta plus, and i get a positive 3 sine squared theta, but one of the other identities we have is a cosine squared theta plus, so i'll write it over here so we remember here, plus a sign squared theta is equal to one.

01:44 So i'm going to separate these out.

01:46 I have a positive three, so here's one...

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