# Right Triangles and Trigonometry

### Terms

### Geometric Mean

The square root of the product of two numbers. That's not so mean, is it?### Pythagorean Triple

A right triangle whose three side lengths are all integers. For instance, a triangle with side lengths of 3, 4, and 5 would be considered a Pythagorean triple.### Pythagorean Theorem

States that in any right triangle, the sum of the lengths of the legs squared equal the square of the length of the hypotenuse. More simply,*a*

^{2}+

*b*

^{2}=

*c*

^{2}.

### 45-45-90 Triangle

A special right triangle where the angles are 45°, 45°, and 90°. In these triangles, both legs are the same length and the hypotenuse equals the length of the leg times the square root of 2. As was written:*One side to find them all…*

### 30-60-90 Triangle

A special right triangle where the angles are 30°, 60°, and 90°. In these triangles, the hypotenuse is equal to 2 times the shortest leg, and the long leg equals the shortest leg times the square root of 3.### Trigonometry

The use of a triangle's angles and side lengths to calculate its angles and side lengths. It seems kind of redundant, but it's not.### Trigonometric Ratios

Specific fractions made by dividing specific side lengths. The three main ones to remember are sine, cosine, and tangent. The**sine**of an angle is the side length opposite the angle over the hypotenuse. The

**cosine**is the adjacent side over the hypotenuse. The

**tangent**is the opposite side over the adjacent side. Capisce?

### Law Of Sines

A relationship stating that in any non-right triangle, the sine of any angle divided by its opposite side equals the sine of another angle divided by its opposite side. In other words, .### Solving A Triangle

Finding all the side lengths and angles in a triangle. At least there's only three of each.### SOHCAHTOA

What you should do after stubbing your toe. Also, a mnemonic for remembering the three main trig ratios: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. Now go get some ice, because that thing's gonna bruise.### Law Of Cosines

A relationship between the angles and side lengths of a triangle, stating that the square of any side length equals the sum of the squares of the two other side lengths minus two times the product of the two other side lengths and the cosine of the opposite angle. Basically, this:*a*

^{2}=

*b*

^{2}+

*c*

^{2}– 2

*bc*cos

*A*.

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