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

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

Key Concepts

Reference Angle

The reference angle is the acute angle that a given angle makes with the horizontal axis. It is used to determine other angles whose trigonometric function values are the same in magnitude but may differ in sign, depending on the quadrant. This concept is essential when finding all angles that satisfy an equation involving the sine function.

Double-Angle Equations

A double-angle equation features a trigonometric function with an angle that is a multiple of the variable, such as sin(2?). Solving such equations typically involves first finding the solutions for the inner angle (2?) and then dividing by the multiple to obtain the solutions for ?, while also carefully adjusting the domain accordingly.

Sine Function and Its Inverse

The sine function relates an angle to the ratio of the opposite side to the hypotenuse in a right triangle. To solve equations like sin(x) = a, the inverse sine function (arcsin) is used to obtain a principal value. However, because the sine function is periodic and has symmetry, additional solutions exist that are found by considering its properties in different quadrants.

Trigonometric Equations

Solving trigonometric equations involves finding all angles within a specified domain that satisfy the given equality. This process requires an understanding of the basic properties of trigonometric functions and their periodic nature, allowing one to generate general solutions and then restrict them to the given interval.

Periodicity of Trigonometric Functions

Trigonometric functions repeat their values in regular intervals, with sine having a period of 360° (or 2? radians). This periodicity means that once a solution is identified, an infinite number of solutions exist, spaced at fixed intervals. When solving equations over a restricted domain, one must take these repeats into account to include all valid solutions.

Transcript

00:01 Okay, so this problem, you need to make sure with your calculator you're in degree mode.

00:06 That's by hitting mode, and so you hit the mode key and go down.

00:12 Make sure that degree is highlighted.

00:16 So whenever they give you sign of something, to undo that is to do inverse sign on both sides.

00:25 So what we have is 2 theta is equal to inverse sign of this value.

00:30 And when you type that in, your calculator will tell you that that answer is negative 51 .656 degrees.

00:40 All right.

00:42 So what does that mean? is in the unit circle, or i guess, just saying that, hey, we have a negative angle.

00:51 Negative angle.

00:52 So we're going clockwise right here.

00:55 So what you want to do is add 360 degrees to this so that you're going.

01:01 Clockwise because that's a full revolution and when i add 360 to that i get that two theta is equal to 308 point i think they asked you to round the nearest 10th or two decimal places sorry but i'm not ready to round yet because i still need to solve for theta and that's by dividing both sides by two so this is one answer that theta is equal to whatever that is divided by two 154 .17 degrees.

01:37 All right, but that's not the only answer because sign is also negative over in this quadrant.

01:44 So you'd have to think about how this angle is also 51 .656 degrees.

01:50 That's gotten from right here...

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