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The Tangent Function, a la Shmoop.

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Archduke James XIV is a tanned gentleman who enjoys spending his days on the beach.

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Unfortunately, James has an overwhelming fear of being the victim of seagull air raids.

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He figures that as long as he can calculate how high the seagulls are flying,

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he can get out of the way in time.

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How high are the seagulls flying?

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We can define the height of the seagulls with the variable h.

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We also know that James is sunbathing exactly 42 feet away from the shoreline.

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Luckily, James has always aspired to be a sailor, so he happens to have his handy-dandy

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sextant, which he can use to tell the angle between two things.

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His sextant tells him the angle of elevation,

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or the angle between the ground and the seagull, is exactly 51 degrees.

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Because we know this is a right triangle with a 90 degree angle, we can use trigonometry

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to help James find h, the height of the seagulls.

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Let's use the tangent function, which tells us the ratio of the opposite side

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to the adjacent side in a right triangle with a given angle.

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So, using what we know, we can plug in the value of 51 degrees for the angle,

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and 42 as the adjacent side.

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We can plug in h as the opposite side, because it is "opposite" the angle.

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To isolate h, we can multiply 42 to both sides,

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to get h equals 42 times the tangent of 51 degrees...

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…to get that the height equals about 52 feet.

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Now that James knows how high the seagulls are flying, he doesn’t have to worry about

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getting nailed by seagull bombs. But James is an adventurous soul, and wants

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to be prepared for other beaches as well.

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So he prepares some other handy tangent values

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to calculate how high seagulls would be flying over neighboring beaches at other angles.

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James decides to find the tangent values of a couple special angles:

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zero degrees, thirty degrees, forty-five degrees, and sixty degrees.

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Using his calculator, James finds that the tangent of zero degrees is zero…

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…the tangent of thirty degrees is one over the square root of three, or .577…

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…the tangent of 45 degrees is 1…

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…and the tangent of 60 degrees is the square root of three, or 1.732.

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James is finally ready to conquer any seagulls, at any beach, any time.

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Now he just has to remember his sunscreen.

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