FUNCTIONS

BY: Chambria Rogers

&

Marc

TRIGONMETRIC RATIO

• The six trigonometric functions are traditionally defined in terms of the six ratios of the sides of a right triangle. This approach is valid for positive angles of measure smaller than 90∘.

• Consider a right triangle △ABC with the angle C being the 90∘ angle.

There are six functions that are the core of trigonometry.

There are three primary ones that you need to understand completely:

• Sine (sin)

• Cosine (cos)

• Tangent (tan)

The other three are not used as often and can be derived from the three primary functions. Because they can easily be derived, calculators and spreadsheets do not usually have them.

• Secant (sec)

• Cosecant (csc)

• Cotangent (cot)

Sine (SIN) FUNCTION

• The law of sines for plane triangles was known to Ptolemy and by the tenth century Abu'l Wefa had clearly expounded the spherical law of sines. It seems that the term

"law of sines" was applied sometime near 1850

• In a right triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse. • The sine function has a number of properties that result from it being periodic and odd.

• The Law of Sines relates various sides and angles of an arbitrary (not necessarily right) triangle:

• sin(A)/a = sin(B)/b = sin(C)/c =

2r.

• where A, B, and C are the angles opposite sides a, b, and c respectively. Furthermore, r is the radius of the circle circumscribed in that triangle.

COSINE

In a right triangle, the cosine of an angle is:

The length of the adjacent side divided by the length of the hypotenuse.

The law of cosines is best thought of as an extension of the Pythagorean Theorem, with a term that adjusts if the included angle is not a right angle. The usual statement of the theorem is descibed in terms of sides a, b, and c; and opposite angles A, B, and C. The usual expression is c2=a2+b2-2abCos(C). The theorem is cyclic about any of the three sides and so it can also be expressed in the alternate forms a2=b2+c2-2bcCos(A) and b2=a2+c2-2acCos(B). Since the cosine of a right angle is zero, each of the equations reduces to the usual form of the Pythagorean

Theorem when the associated angle is 90o.

TANGENT statement about the relationship between the tangents of two angles of a triangle and the