TRIGONOMETRY

Areas of focus:

One revolution is 360o, and is also 2 radians. Thus, due to linear proportionality of the two scales, the conversion from x degrees to y radians is:

One radian is equal to 3.14159..., and is normally approximated by 3.14. The following table gives equivalent angles in degrees, radians, and revolutions.

 Degrees Radians Revolutions 0o 0 0 30o 45o 60o 90o 180o 270o 360o 1

The trigonometric functions are named sine, cosine, tangent, cotangent, secant, and cosecant. A trigonometric function has one argument that is an angle and will be denoted "". In writing the trigonometric functions one uses the abbreviated forms: , , , , , and , respectively. Also, sometimes these are written as , , , , , and , respectively.

The value of each trigonometric function for an acute angle (<90o) can be directly related to the sides of a right triangle. Consider the angle in the following figure. The values of the trigonometric functions for this angle are given as:

Note: the exponents of trigonometric functions follow a special rule. If the exponent "n" is positive, then one writes in place of . For example,

The same rule does not apply to negative exponents since the exponent "-1" is reserved for the inverse trigonometric function.

Functions of Complementary Angles:

In this figure, and are complementary angles, meaning . Examination of the basic relation between the trigonometric functions and the sides of the triangle reveal the following relations between the complementary angles and.

Since , we can also write:

Pythagorean Theorem:

The Pythagorean theorem states that for a right triangle, as shown, there exists a relation between the length of the sides given be

a2 + b2 = c2

There are also Pythagorean triples for (a,b,c), such as (3,4,5), (5,12,13) and (7,24,25) sided triangles, and all constant multiples of these triplets (e.g., (6,8,10)).

Fundamental Relations Among Trigonometric Functions:

From the Pythagorean Theorem of plane geometry we know that x2 + y2 = r2. This can be used to derive a basic relation between the sine and cosine functions.