ShmoopTube

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Tangent Half Angle Formula, a la Shmoop. Vido the Mosquito is exhausted after returning

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from his business trip to Trigonometry Springs. Vido desperately needs an espresso. At Jolts

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and Volts Coffee he orders his much needed jolt of caffeine.

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As Vido digs in his pockets for theÉtangent of pi over eightÉ dollars he needs to pay

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for his double shotÉ Éhe realizes he is in Half Angle-os and he

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only has Trig Springs money. In Half Angle-os, all money is converted using

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the following formula: Tangent of u over 2 equals one minus the cosine

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of u over sine of u. How will Vido convert his money to the correct

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amount for his much needed coffee? First of all we can see that tangent of pi

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over 8 can be rewritten as ? times pi over 4.

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If we do this, we can get rid of the denominator 2 in our formula.

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Then, if we substitute tangent of pi over 4 in for u, we get the following:

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1 minus cosine of pi over 4 all divided by sine of pi over 4.

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Since cosine of pi over 4 equals the square root of 2 over 2, and the sine of pi over

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4 equals the square root of 2 over 2É Éour formula becomes:

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1 minus the square root of 2 over 2 all over square root of 2 over 2.

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Having too many terms just brings troubleÉ Éso first, to simplify the numerator, we

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can put both terms under the same common denominator. Let's do this by multiplying 1 by 2 over 2,

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to get 2 over 2 minus root 2 over 2. We subtract fractions by subtracting across

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the numerator of the fractions and keeping the denominator the same, so we get 2 minus

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root 2 over 2 on the top of the bigger fraction. We hate fractions in the denominator, so letÕs

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multiply the numerator, 2 minus root 2 over 2É by the reciprocal of the denominator to

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get rid of it. The 2 in the numerator and denominator cancel

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out, so we're left with 2 minus root 2 divided by root 2.

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We canÕt have radicals in the denominator, so we will rationalize it by multiplying the

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fraction by root 2 over root 2. This gives us the fraction root 2 times the

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quantity 2 minus root 2 all over 2. To simplify, letÕs distribute the root 2

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across the quantity 2 minus root 2 to get 2 root 2 minus 2 all divided by 2.

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Since we have a 2 on top and a 2 on bottom, they cancel each other out, so our final quantity

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is root 2 minus 1. Fed up with having to go through so much math,

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Vido throws his money at the barista and demands his coffee.

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The barista calmly reaches back and points to a sign saying ÒWe reserve the right to

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refuse service to anyone.Ó Maybe try the Starbugs across the street,

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Vido.

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