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

Trigonometric Identities

Trigonometric identities, such as the Pythagorean identity, allow you to express one trigonometric function in terms of another. This is particularly useful in simplifying and transforming equations—in this case, rewriting cos²? in terms of sin²? to obtain an equation with a single trigonometric function.

Quadratic Equations in Trigonometric Functions

When a trigonometric equation can be rewritten in a quadratic form (for example, in sin ?), standard algebraic techniques like factoring or the quadratic formula can be employed to find the solutions. This method transforms a trigonometric problem into an algebraic one.

Solution Domain Considerations

Trigonometric equations are periodic, but frequently problems restrict the solutions to a specific interval (such as 0° to 360°). It is crucial to consider these domain restrictions when selecting the valid solutions from the general set of roots provided by the quadratic equation.

Standard Angle Values

Having an understanding of the specific angles at which trigonometric functions achieve well-known values enables you to determine the exact solutions. This knowledge is critical when matching algebraic results (like specific sine or cosine values) with their corresponding angles.

Transcript

00:01 Just solve the equation 3 minus 3, sine theta, minus 2, cosine squared data, equals 0.

00:06 We have cosine squared here, which we can substitute using the pythagorean identity.

00:14 So we'll bring down the 3 minus 3, sine theta, minus 2, and then we'll substitute for cosine squared theta, 1 minus sine squared theta, and we'll set that equal to 0.

00:31 We're going to distribute that negative 2.

00:33 So we have 3 minus 3 sine theta minus 2 plus 2 sine square theta equals 0.

00:46 So we'll do a little bit of rearranging, a little bit of combining like terms.

00:50 So we'll start with 2 sine square theta.

00:55 Then we'll bring the minus 3 sine theta and then we have like terms of three minus two so that's going to give us plus one and we'll set that equal to zero so now we're going to factor this we have two sine squared theta so we know we need a two sign theta and a sign theta we know we need to multiply out to one so we need a one and one and then we need to also add up to a negative three so to add up to a negative, multiply to a positive, we need a minus sign, minus sign.

01:37 So there are two equations we're going to solve.

01:39 We're going to solve 2 sine theta minus 1 equals 0.

01:46 And we're going to solve sine theta minus 1 equals 0.

01:51 So on this side, we'll add 1.

01:54 So we have 2 sine theta equals 1, divide by 2...

Please give Ace some feedback