Solve the given trigonometric equation on 0^∘ ≤θ<360^∘ and express the answer in degrees to two

Solve the given trigonometric equation on 0^∘ ≤θ<360^∘ and...

Key Concepts

Trigonometric Equations

This concept involves equations that include trigonometric functions such as sine, cosine, and tangent. Solving these equations often requires algebraic manipulation, the use of trigonometric identities, and an understanding of the periodic nature of these functions.

Quadratic Equations in Trigonometric Functions

Many trigonometric equations can be rewritten in a quadratic form by substituting a trigonometric expression with a variable. This allows the use of methods from algebra, such as factoring or the quadratic formula, to solve the equation before converting back to the trigonometric variable.

Double-Angle Identities

Double-angle identities are specific trigonometric formulas that simplify expressions involving functions of double angles, such as 2?. They are crucial in rewriting and solving equations where functions of double angles appear, enabling the reduction of the equation to a simpler form.

Periodicity in Trigonometric Equations

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. When solving trigonometric equations, it is important to account for all possible solutions within a given interval by adding integer multiples of the period, ensuring that the full set of solutions is identified.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angle that corresponds to a given trigonometric value. These functions are essential in the final steps of solving trigonometric equations, as they allow conversion from function values back to angle measures.

Transcript

00:01 All right.

00:02 I don't know how prepared i am for this problem, but i'm going to jump right into it plus 2 sine of theta, excuse me, and then minus 2.

00:25 There you go.

00:26 Now i believe i have the right end problem.

00:28 There we go.

00:29 So just off to the side, i like to think about 15 times negative 2 is negative 30.

00:34 And i can think of factors of negative 30 that add to be 1.

00:39 It'll be 6 and negative 5.

00:42 So how i would get this problem is now i'm going to factor.

00:49 And i'm just thinking about, okay, well, this five is going to be complicated to get because i can't get a five from this negative two.

00:56 But i can if i had five sign of two theta and over here a three sign of two theta.

01:05 That gives me 15, what i want.

01:07 And as i go to foil this, you know, i want five and want it to be negative five.

01:13 So i'm going to make this negative 1.

01:15 And on the inside here, well, three times 2 gives me that 6.

01:21 And i want it to be a positive term.

01:23 And i can confirm that that's right because 2 times negative 1 is negative 2 from up here.

01:27 So now i can use a zero product property because if the left side is equal to negative 2 -fis, then now it equals 0.

01:40 0 times anything is that thing.

01:42 And then over here, sign of 2 theta could equal 1 third.

01:49 And then from here, holy cow, this is going to be a lot of work.

01:52 You want to do the inverse sign of negative 2fis.

01:59 This is going to be a lot of work because this isn't even the right answer.

02:03 Negative 23 .5, 7, 8 degrees, because that's in the fourth quadrant.

02:09 Man, it's smaller than zero.

02:11 I don't want that answer.

02:14 Okay.

02:16 Well, i started it, so let's finish it.

02:18 So i would need to add 360 to that...

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