Let ACB be a central angle in a unit circle (See figure 1), the radian measure q of the angle ACB is defined to be the length of the circular arc AB.

FACT: p radians = 180^o

Here are three special triangles that you should commit to memory. Know them, use them, they are your friends!!J

figure 1

DEFINITION:

An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies along the positive x-axis.

FACT:

Angles measured counterclockwise from the positive x-axis are assigned positive measures. Angles measured clockwise are assigned negative values.

There is a useful relationship between the length of S of an arc ABC on a circle of radius r and radian measure q of the angle and the arc subtends at the circle's center C. The relationship is

This is the arc length formula.

EXAMPLE 1:

An arc of length 26 subtends a central angle of a circle of radius 4p . Find the angle's radian measure.

SOLUTION:

If we place this triangle on a circle with radius r (r will be the hypotenuse and the adjacent side will be the positive x-axis (see the following figure)), then the trig functions will be defined as follows.

FACT:

Any point P (x, y) can be written in terms of (r, q ).

P (x, y) = P (r cos q , r sin q )

The following table contains values of the six trig functions at specific angles. You should commit these values to memory. We will be using them throughout this course and any other math course that you will take beyond calculus I. They will also be used in physics, engineering and other related science fields.

DEGREES

0°

30°

45°

60°

90°

180°

270°

RADIANS

0

p

SIN q

0

1

0

-1

COS q

1

0

-1

0

TAN q

0

1

UND

0

UND

COT q

UND

1

0

UND

0

SEC q

1

2

UND

-1

UND

CSC q

UND

2

1

UND

-1

Positive

Negative

sin q

I, II

III, IV

cos q

I, IV

II, III

tan q

I, III

II, IV

cot q

I, III

II, IV

sec q

I, IV

II, III

csc q

I, II

III, IV

y = sin x

D: (-¥ , ¥ )

R: [-1, 1]

Period: 2p

y = cos x

D: (-¥ , ¥ )

R: [-1, 1]

Period: 2p

y = tan x

R: (-¥ , ¥ )

Period: p

y = cot x

D: {x | x ¹ ± np }

R: (-¥ , ¥ )

Period: p

y = sec x

R: (-¥ , -1] È [1, ¥ )

Period: 2p

y = csc x

D: {x | x ¹ ± np }

R: (-¥ , -1] È [1, ¥ )

Period: 2p

FACT:

Cosine and secant are even functions.

FACT:

Sine, cosecant, tangent, and cotangent are odd functions.

These are the identities that we will use most often. Therefore, commit them to memory.

sin ² x + cos ² x = 1

1 + tan ² x = sec ² x

1 + cot ² x = csc ² x

ANGLE SUM FORMULAS

cos (A + B) = cos A cos B - sin A sin B

sin (A + B) = sin A cos B + cos A sin B

DOUBLE ANGLE FORMULAS AND THEIR MODIFICATIONS

cos 2q = cos ² q - sin ² q

sin 2q = 2sin q cos q