Chapter 4 Vocabulary

Section 4.1 vocabulary

An angle is determined by a rotating ray (half-line) about its endpoint.

The starting point of the ray is the initial side of the angle.

The position of the ray after the rotation is the terminal side of the angle.

The endpoint of the ray is the vertex of the angle.

When an angle fits a coordinate system in which the origin is the vertex of the angle, and the initial side coincides with the positive x-axis that angle is in standard position.

Positive angles are generated by counterclockwise rotation.

Negative angles are generated by a clockwise rotation.

Two angles that have the same initial and terminal sides are called coterminal angles.

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.

A central angle is an angle whose vertex is the center of the circle.

One radian is the measure of a central angle Ѳ that intercepts an arc s equal in length to the radius r of the circle.

Ѳ = s / r , where Ѳ is measured in radians

Angles between 0 and ∏ / 2 are called acute angles.

Angles between ∏/2 and ∏ are called obtuse angles.

A way to measure angles is in degrees where 1 degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex.

Complementary angles add to be 90 degrees or∏/2.

Supplementary angles sum to equal 180 degrees or ∏.

Linear speed

Linear speed = Arc length / time

Angular speed

Angular speed = central angle/ time

The unit circle Given by the equation : X2 + y2 = 1

Definitions of Trigonometric functions

• Sin (t) = y• Cos(t) = x• Tan(t) = y/x• Csc(t) = 1/y• Sec(t) = 1/x• Cot(t) = x/y

• A function f is periodic if there exists a positive real number c such that :

f(t + c) = f(t) For all t in the domain of f.The least number c for which f is

periodic is called the period of f.

Even/ odd trig functions

Even • cos(-t) = cos(t)• sec(-t) = sec(t) odd• Sin(-t) = -sin(t) • tan(-t) = -tan(t) • csc(-t) = -csc(t) • cot(-t) = -cot(t)

Section 4.3 Vocabulary

Right triangle def. of Trig Functions

• Sin(Ѳ) = opp/hyp• Cos(Ѳ)= adj/hyp• Tan(Ѳ) = opp/adj• Csc(Ѳ) = hyp/opp• Sec(Ѳ) = hyp/adj• Cot(Ѳ) = adj/opp

Sines of special angles

•Sin(30) =sin(∏/6) = ½•Sin (45) = sin(∏/4) = √2/2•Sin(60) = sin(∏/3) = √3/2

Cosines of special angles

•Cos(30) = cos(∏/6) = √3/2•Cos(45) = cos(∏/4) = √2/2•Cos(60) = cos(∏/3) = ½

Tangents of Special angles

• Tan(30) = tan(∏/6) = √3/3• Tan(45) = tan(∏/4) = 1• Tan(60) = tan(∏/3) = √3

Reciprocal Identities• Sin(Ѳ) = 1/csc(Ѳ)• Cos(Ѳ) = 1/ sec(Ѳ)• Tan(Ѳ) = 1/cot(Ѳ)• Csc(Ѳ) = 1/sin(Ѳ)• Sec(Ѳ) = 1/cos(Ѳ)• Cot(Ѳ) = 1/tan(Ѳ)

Quotient identities

• Tan(Ѳ) = sin(Ѳ) / cos(Ѳ)

• Cot(Ѳ) = cos(Ѳ) / sin(Ѳ)

Pythagorean Identities

•Sin2(Ѳ) + cos2(Ѳ) = 1•1 + tan2(Ѳ) = sec2(Ѳ)•1 + cot2(Ѳ) = csc2(Ѳ)

Angle of elevation

•The angle from the horizontal up to the object

Angle of Depression

•The angle from the horizontal downward to the object.

Section 4.4 Vocabulary

Definitions of Trig Functions

Sin Ѳ = y/rcos Ѳ = x/rTan Ѳ = y/xCot Ѳ = x/ySec Ѳ = r/xCsc Ѳ = r/y

Reference Angle• Let Ѳ be an angle in standard

position. Its reference angle is the acute angle Ѳ’ formed by the terminal side of V and the horizontal axis.

Section 4.6 Vocabulary

Amplitude • The amplitude of y = a sin(x) And y = a cos(x) Represents half of the distance

between the max and the min values of the function, and is given by

Amplitude = |a|

Period• The b be a positive real

number. The period of y = a sin(bx) and t = a cos(bx) is given by

Period = 2∏/b

Damping factor•In the function f(x) = x sin(x), the factor x is called the damping factor.

Section 4.7 Vocabulary4

Inverse sine functiony = sin (x) has a unique inverse

function called inverse sine function. It is denoted by

Y =arcsin(x) or y = sin-1 (x)

Inverse cosine functiony = cos (x) has a unique inverse

function called inverse cosine function. It is denoted by

Y =arccos(x) or y = cos-1 (x)

Inverse tangent functiony = tan (x) has a unique inverse

function called inverse tangent function. It is denoted by

Y =arctan(x) or y = tan-1 (x)