Question Video: Using Inverse Trigonometric Functions to Solve Trigonometric Equations Mathematics

Find 𝑚∠𝐸 given tan 𝐸 = 18.5845 and ∠𝐸 is an acute angle. Give your answer to the nearest second.

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Video Transcript

Find the measure of angle 𝐸 given that the tan of 𝐸 equals 18.5845 and angle 𝐸 is an acute angle. Give your answer to the nearest second.

Since we know that angle 𝐸 is an acute angle, we know that we’ll only be dealing with values in the first quadrant. We know that the tan of 𝐸 is equal to 18.5845. We know that the tangent of some angle equals a ratio of side lengths. And we also know that if we’re given a ratio of side lengths for some tangent, using the inverse tangent, we can find the angle measure. And that means we’ll want to take the inverse tangent of both sides of this equation. The inverse tan of tan of 𝐸 equals 𝐸. And before we find the inverse tan of 18.5845, we need to think about “are we dealing with radians or degrees?” We need to round this value to the nearest second. A second is a fraction of minutes, and a minute is a fraction of a degree. That means we’ll need to operate this inverse tangent in degrees.

Making sure our calculator is set to degrees, we get that 𝐸 equals 86.91998286 continuing degrees. This means 𝐸 has 86 whole degrees and another partial degree. In order for to us to turn this partial degree into minutes and seconds, we have to remember that one degree equals 60 minutes. And that means to find out how many minutes 0.91998286 continuing degrees would be, we multiply by 60. And when we do that, we get 55.19897132 continuing minutes, 55 whole minutes and a partial minute.

Our final step will be converting this partial minute into seconds. And for that, we have to know that one minute equals 60 seconds. So, we have 86 degrees, 55 minutes and then we’ll have 0.19897132 times 60 minutes, which gives us 11.93827 continuing seconds. To round to the nearest second, we’ll round that 11.9 up to 12. And we’ll say that the measure of angle 𝐸 is equal to 86 degrees, 55 minutes, and 12 seconds.

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