Find all values of θ, if θ is in the interval [0°, 360°) and has ...

Question

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value.

sec θ = -2√3
               3

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine all possible values of theta in the interval from 0 to 360 degrees inclusive of zero but not of 360 degrees, which have the same value as the given function value. Our given function value is the Cosic can of theta equals negative square two. Our answer choices are answer choice. A theta equals 45 degrees and 225 degrees. Answer choice B theta equals 135 degrees and 225 degrees. Answer choice C theta equals 135 degrees and 315 degrees. And answer choice D theta equals 225 degrees and 315 degrees. All right. So we take a look and we have this exact value here for a Cosic can at negative square root two. And what we can do is look at our textbook charts and recall from previous lessons, where is the Cosic can of an angle equal to the square rut of two? And it's true that the Cosic can of theta is equal to a positive square up of two. When theta equals degrees. But we're given a negative square it of two for our exact value. So we need to figure out in what quadrants the coy can is negative. And then we need to figure out what our angles are if we have a reference angle of 45 degrees. So let's draw a quick sketch of a coordinate plane. We have a vertical y axis and a horizontal X axis that come together at the origin in the middle. And so we recall from previous lessons where different trigonometric functions are positive. Using that pneumonic device. All students take calculus in the first quadrant up into the right of the origin. All of the trigonometric functions are positive, but we're looking for negative ones. So we can rule out quadrant one in the second quadrant up into the left of the origin, the sign function and it's reciprocal the co can are positive. So the co is also positive in the second quadrant, but we're trying to find where it's negative. So we can eliminate the second quadrant. That means we're talking about the third quadrant down into the left of the origin and the fourth quadrant down into the right of the origin. So our reference angle is that 45 degree angle in the first quadrant, it's got its initial side along the positive part of the X axis and its terminal side is in the first quadrant directly between the positive part of the X axis and the positive part of the y axis. And so what is the angle in quadrant three? That would have a reference angle 45 degrees? And what is the angle in quadrant four that would have a reference angle of 45 degrees? And in order to find the third quadrant angle, we are going to take 180 degrees plus our reference angle 45 degrees and 180 degrees plus 45 degrees is 225 degrees. So there's one measurement of theta that is true for the coy can being equal to negative square root two. Now, to find our quadrant four angle that has a reference angle of 45 degrees, we're going to take 360 degrees minus our reference angle 45 degrees and 360 degrees minus 45 degrees is 315 degrees. There's our other value for theta that makes this statement true. So we plug in 225 degrees and 350 degrees in for and we get the negative square root of two. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. sec θ = -2√3 3

Key Concepts

Secant Function

Quadrants of the Unit Circle

Inverse Trigonometric Functions

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