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

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

Key Concepts

General Solutions of Sine Equations

When solving an equation of the form sin A = sin B, one uses the general solution formula: A = B + 360°·k or A = 180° – B + 360°·k, where k is an integer. This approach accounts for the periodic nature of sine and provides all possible solutions before applying any domain restrictions.

Periodicity and Domain Restrictions

Trigonometric functions are periodic, meaning their values repeat at regular intervals. In solving equations, it is important to consider the periodicity to generate all potential solutions. Once the general solutions are obtained, the next step involves applying the given domain restrictions (for example, 0° ? ? ? 360°) to determine which solutions are valid within the specified interval.

Trigonometric Equations

This concept involves solving equations that contain trigonometric functions. It requires using algebraic techniques and trigonometric identities to manipulate and simplify the expressions, ultimately finding the angle values (or sets of angles) that satisfy the given equation within a specific interval.

Cofunction Identities

Cofunction identities are relationships between pairs of trigonometric functions where one function of an angle is equal to its cofunction of the complementary angle. For example, sin x equals cos(90° – x) in degree measure. This identity is especially useful when an equation includes both sine and cosine functions, as it allows one function to be expressed in terms of the other, thereby simplifying the problem.

Transcript

00:01 This question that we're given here is asking for us to find the values of theta between 0 and 360 for this equation here.

00:12 And we're going to use one of the identities we're provided.

00:15 And in this case, it's the half angle identities.

00:18 And so for sign of one half theta, we're going to replace that with plus and minus the square root of one minus the cosine theta over two is equal to.

00:31 To cosine theta.

00:33 So now, in order to solve this, we're going to square both sides.

00:38 I guess i should put that on the other side.

00:40 There we go.

00:41 We square that.

00:42 We're going to square that.

00:43 And we end up with 1 minus cosine theta over 2 is equal to cosine squared theta.

00:53 And then i'm just going to multiply both sides by two.

00:57 And then i get one minus cosine theta is equal to two cosine squared theta.

01:04 And then let's get everything onto one side.

01:07 I always like to keep the number in front of the squared term positive.

01:11 So we'll move everything over there to the left by subtracting one and adding a cosine theta.

01:17 So we get two cosine squared theta plus a cosine theta, plus a cosine theta, minus 1...

Please give Ace some feedback