# Tangent Half-Angle Formula

Mosquitos need their caffeine, just like the rest of us. Wouldn't think zipping around and stinging people all day long sap *your* energy?

ACT Mathematics | Trigonometry |

Circles | Tangents and Secants Understand and apply theorems about circles |

Functions | Prove and apply trigonometric identities |

Geometry | Right Triangles and Trigonometry Understand and apply theorems about circles |

Language | English Language |

Math | Algebra II |

Trigonometry | Modeling using trigonometric functions Use of trigonometric identities |

### Transcript

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Vido.