**TRIGONOMETRY**

** **

**Areas of focus:**

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- Degrees versus radians
- Trigonometric functions
- Trigonometric relations between complementary angles
- Pythagorean Theorem
- The fundamental relation between sine and cosine
- The unit circle and visualizing the trigonometric functions
- Inverse of trigonometric functions
- Law of sines
- Law of cosines
- Values of trigonometric at specific angles
- Trigonometric identities
- Curves of sine, cosine, and tangent
- Approximation for small angles: sine, cosine, and tangent

One revolution is 360^{o}, 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 |

0 |
0 |
0 |

30 |
||

45 |
||

60 |
||

90 |
||

180 |
||

270 |
||

360 |
1 |

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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* (<90^{o}) 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:

** **

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

a^{2} + b^{2} = c^{2}

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 x^{2} +
y^{2} = r^{2}. This can be used to derive a basic relation
between the sine and cosine functions.