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

Question

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. sin θ = 0.52991926

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 from 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 the sign of theta equals 0.74314482. Our answer choices are answer choice. A 44 degrees and 136 degrees. Answer choice B 48 degrees and 132 degrees. Answer choice C 52 degrees and 128 degrees and answer choice D 63 degrees and degrees. All right, we're trying to find values of theta. And when we have a trigonometric function of an angle equal to a number, we can use the inverse function buttons on our calculator. So because we have sign here, we want to use the inverse sign button and that's going to look like it says sign to the negative one. You want to make sure that you are in degrees when you do this and then you hit that inverse sign button and then input our 0.74314482. And when you hit enter, you should get 47.99999953. And because we get to round to the nearest degree that rounds to 48 degrees, so there's one solution for theta, but we need another one and we need it between zero and 360 degrees. So how do we figure that out? Well, if we envision a coordinate plane, we have a vertical Y axis and a horizontal X axis, they come together at the origin in the middle, where is 48 degrees? Well, that's going to be in quadrant one, we have our vertex at the origin initial side along the positive part of the X axis and then 48 degrees, the terminal side is heading toward positive infinity for X and Y in the first quadrant. And we know that sign is equal to positive values in the first quadrant. All of the trigonometric functions are equal to positive values in the first quadrant. But where else is sine equal to positive values? We recall from previous lessons that that's also going to be in quadrant two. So if we use 48 degrees as our reference angle to figure out what that would be in quadrant two, we're going to take 180 degrees minus our 48 degrees and when we do that, we get 132 degrees, there is our second quadrant angle. That is also true. But will the sign of that 132 degrees will also equal 0.74 and so on. We also know this and this is summed up in the statement that the sign of X that would be our degrees equals the sign of degrees minus X. All right, we look at our answer choices and this matches with answer choice. B 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. sin θ = 0.52991926

Key Concepts

Inverse Sine Function (Arcsin)

Sine Function Symmetry and Periodicity

Reference Angle and Quadrant Determination

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