Find two angles in the interval [0°, 360°) that satisfy each of t...

Question

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree.

tan θ = 0.70020753

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine two values of theta that satisfy the given statement consider values that are in the interval of zero degrees to 360 degrees, inclusive of zero degrees, but not of 360 degrees. Round your answers to the nearest degree. Our given statement is that the tangent of theta equals 1.60033453. Our answer choices are answer choice. A 58 degrees and 238 degrees. Answer choice B 76 degrees and 256 degrees. Answer choice C 58 degrees and 122 degrees and answer choice D 76 degrees and 104 degrees. All right. So we're trying to figure out the value of theta. And when we have a trigonometric function of an angle equal to a value, we can use our calculators and we want to use the inverse function button. So because we have tangent here, we're going to use the inverse tangent button. That's the tangent raised to the negative one. And then you input our number from the other side of the equal sign that 1.60033453. And now this part is very important, you need to have your mode of your calculator in degrees. If you do, you'll get an answer of 58.00000002. And because we get to round to the nearest degree that rounds to 58 degrees. So that's one of our values of an, of our angle. Now, what's another one in between zero and 360 degrees? There's a couple ways to think about this. We can envision a coordinate plane. We've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. Where's our 58 degrees? That's going to be in the first quadrant. So we've got our vertex at the origin, the initial side heading toward positive infinity along the X axis and our terminal side is in the first quadrant heading toward positive infinity. For both X and Y, there's 58 degrees in quadrant one. Now, tangent is positive in the first quadrant, we've got that tangent of theta equals 1.6. That's a positive number. But where else is tangent going to be positive in between zero and 360 degrees? It's also going to be positive in the third quadrant, we recall from previous lessons, tangent also positive in the third quadrant. So if we use 58 degrees as our reference angle for quadrant three, we're going to add 180 degrees to that 58 degrees and that will get us our third quadrant angle that makes this statement true for the tangent. So 180 degrees plus 58 degrees equals 238 degrees. There's our quadrant three angle, we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. tan θ = 0.70020753

Key Concepts

Tangent Function

Inverse Trigonometric Functions

Angle Solutions in a Given Interval

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