"Trigon" is Greek for triangle , and "metric" is Greek for measurement. The trigonometric ratios are special measurements of a right triangle (a triangle with one angle measuring 90 ° ). Remember that the two sides of a right triangle which form the right angle are called the legs , and the third side (opposite the right angle) is called the hypotenuse .

There are three basic trigonometric ratios: sine , cosine , and tangent . Given a right triangle, you can find the sine (or cosine, or tangent) of either of the non- 90 ° angles.

sine = lengthofthelegoppositetotheangle lengthofhypotenuse abbreviated"sin" cosine = lengthofthelegadjacenttotheangle lengthofhypotenuse abbreviated"cos" tangent = lengthofthelegoppositetotheangle lengthofthelegadjacenttotheangle abbreviated"tan"




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Example:

Write expressions for the sine, cosine, and tangent of ∠ A .


The length of the leg opposite ∠ A is a . The length of the leg adjacent to ∠ A is b , and the length of the hypotenuse is c .

The sine of the angle is given by the ratio "opposite over hypotenuse." So,

sin ∠ A = a c

The cosine is given by the ratio "adjacent over hypotenuse."

cos ∠ A = b c

The tangent is given by the ratio "opposite over adjacent."

tan ∠ A = a b

Generations of students have used the mnemonic " SOHCAHTOA " to remember which ratio is which. ( S ine: O pposite over H ypotenuse, C osine: A djacent over H ypotenuse, T angent: O pposite over A djacent.)

Other Trigonometric Ratios

The other common trigonometric ratios are:

secant = lengthofhypotenuse lengthofthelegadjacenttotheangle abbreviated"sec" sec ( x ) = 1 cos ( x ) cosecant = lengthofhypotenuse lengthofthelegoppositetotheangle abbreviated"csc" csc ( x ) = 1 sin ( x ) secant = lengthofthelegadjacenttotheangle lengthofthelegoppositetotheangle abbreviated"cot" cot ( x ) = 1 tan ( x )




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Example:

Write expressions for the secant, cosecant, and cotangent of ∠ A .


The length of the leg opposite ∠ A is a . The length of the leg adjacent to ∠ A is b , and the length of the hypotenuse is c .

The secant of the angle is given by the ratio "hypotenuse over adjacent". So,