In 9-14, find, to the nearest tenth of a degree, the values of θin the interval 0^∘ ≤θ<360^∘ tha

In 9-14, find, to the nearest tenth of a degree, the values...

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the inverse cosine, are used to determine angle measures when the value of a trigonometric function is known. They provide principal values, which serve as starting points for finding all solutions within a given interval by incorporating the periodicity and symmetry of the original trigonometric function.

Periodicity and Symmetry in Trigonometry

Trigonometric functions are periodic, meaning their values repeat over regular intervals; for cosine, the period is 360 degrees. Additionally, cosine's even symmetry implies that the cosine of an angle is equal to the cosine of its negative. These properties allow for the determination of all angles that satisfy a given cosine equation within a specified range by accounting for additional solutions beyond the principal value.

Solving Trigonometric Equations

This concept involves manipulating equations that contain trigonometric functions to isolate the trigonometric term and then determine the corresponding angle solutions. In these types of problems, algebraic techniques are first used to simplify the equation, and then the properties of trigonometric functions are applied to find all angles within a specified interval that satisfy the equation.

Properties of the Cosine Function

Understanding the cosine function is key in solving equations where cosine is involved. The cosine function is periodic, even, and its values range between -1 and 1. It exhibits specific behavior in different quadrants of the unit circle, being positive in the first and fourth quadrants and negative in the second and third, which is essential when considering solutions that arise from both positive and negative values of cosine.

Transcript

00:01 For this problem, we can simplify this equation, or factor i've added this equation to 5 cosine theta plus 2 and 5 cosine theta minus 2.

00:15 You could do it something like this because if you know the format x squared minus 1, this can be factored into x plus 1 times x minus 1.

00:25 That's a memorization.

00:27 Or you can also do it by moving negative 4 to the side divided by 25 and then square rooting.

00:35 Because yes, right, and this gives us, after setting both sides of zero, that cosine of theta equals plus or minus two -fits.

00:46 So to get this to degrees, we would need to arc cosine our value.

00:52 So if we were to arc cosine, two -fits, actually we would only need to find the arc percent of two -fits actually with using the calculator because everything else we can do using the unit circle.

01:09 So if we were to arc -sign, arc -cosin, two -fits, and multiply by 180 and divide it by pi, right, because we need to convert, because arc -signing by cosine would only give us this in radiance, mohapho by 180, and divide by pi to converge degrees, would you get that this is equals 66 .4.

01:31 However, if we look at a unicycle, i'm going to draw a representation out one, zero, one, right? and each gap, here represents a different quadrant, starting from one, two, three, four, right? meaning each from, this is zero degrees and nine degrees, there's a difference in nine degrees in between each one.

01:55 We know for a fact that values in the quadrants tend to mirror each other.

02:01 What i mean by that is that if there is some value right here, if you were to flip it over, say this is like a mirror, the 90 degree value, the same value here would be here, although in a different sign, because the only positive values and the cosine is in the first and fourth, and these are negative.

02:25 However, because we're looking for both positive and negative two -fits, we can go run with this value...

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