The Tangent Function
Ever had someone tell you that getting pooped on by a bird was good luck? Yeah, we never fell for that one either. Watch this video to see how math can help you avoid those stinky bird bombs.
|Circles||Tangents and Secants|
|Functions||Extend the domain of trigonometric functions using the unit circle|
|Trigonometry||Modeling using trigonometric functions|
Use of trigonometric identities
How high are the seagulls flying?
We can define the height of the seagulls with the variable h.
We also know that James is sunbathing exactly 42 feet away from the shoreline.
Luckily, James has always aspired to be a sailor, so he happens to have his handy-dandy
sextant, which he can use to tell the angle between two things.
His sextant tells him the angle of elevation,
or the angle between the ground and the seagull, is exactly 51 degrees.
Because we know this is a right triangle with a 90 degree angle, we can use trigonometry
to help James find h, the height of the seagulls.
Let's use the tangent function, which tells us the ratio of the opposite side
to the adjacent side in a right triangle with a given angle.
So, using what we know, we can plug in the value of 51 degrees for the angle,
and 42 as the adjacent side.
We can plug in h as the opposite side, because it is "opposite" the angle.
To isolate h, we can multiply 42 to both sides,
to get h equals 42 times the tangent of 51 degrees...
…to get that the height equals about 52 feet.
Now that James knows how high the seagulls are flying, he doesn’t have to worry about
getting nailed by seagull bombs. But James is an adventurous soul, and wants
to be prepared for other beaches as well.
So he prepares some other handy tangent values
to calculate how high seagulls would be flying over neighboring beaches at other angles.
James decides to find the tangent values of a couple special angles:
zero degrees, thirty degrees, forty-five degrees, and sixty degrees.
Using his calculator, James finds that the tangent of zero degrees is zero…
…the tangent of thirty degrees is one over the square root of three, or .577…
…the tangent of 45 degrees is 1…
…and the tangent of 60 degrees is the square root of three, or 1.732.
James is finally ready to conquer any seagulls, at any beach, any time.
Now he just has to remember his sunscreen.