Chapter 4- Trigonometric Functions

unit circle

A circle of radius 1 centered at the origin of a coordinate system.

periodic function

A function with values that repeat at regular intervals. There exists a positive real number c such that f(t+c)=f(t) for all values of t in the domain of f.

midline

A horizontal axis that is the reference line about which the graph of a sinusoidal function oscillates.

oblique triangles

A triangle that is not a right triangle.

circular function

A trigonometric function defined as a function of the real number system using the unit circle.

radian

A uit of angular measurement equal to 180 degrees/pi.

damped wave

A wave whose amplitude decreases, such as the graph of a damped trigonometric function.

quadrantal angle

An angle in standard position that has a terminal side that lies on one of the coordinate axes.

coterminal angles

Angles in standard position that have the same initial and terminal sides, but different measures.

sinusoid

Any transformation of a sine function. The general forms of sinusoidal functions are y = a sin (bx+c) + d and y = a cos (bx+c) + d, where a, b, c, and d are constants and neither a nor b is 0.

period

For a function y=f(t), the smallest positive number c for which f(t+c)=f(t).

vertical shift

For a sinusoidal function, a vertical translation that is the average of the maximum and minimum values of the function.

phase shift

For a sinusoidal function, the difference between the horizontal position of a function and that of an otherwise similar sinusoidal function.

frequency

For a sinusoidal function, the number of cycles the function completes in a one unit interval. The frequency is the reciprocal of the period.

amplitude

Half the distance between the maximum and minimum values of a sinusoidal function.

Law of Cosines

If triangle ABC has side lengths a, b, and c, For any triangle with sides of lengths a, b, and c, and with C the angle opposite the side with length c, c^2 = a^2 + b^2 - 2ab(CosC).

sector

In a circle, the region bounded by a central angle and its intercepted arc.

damping factor

In a damped trigonometric function of the form y = f(x) sin bx or y = f(x) cos bx, f(x) is the damping factor.

secant

In a right triangle with acute angle theta, the ratio comparing the length of the hypotenuse to the side adjacent to theta. It is the reciprocal of the cosine ratio.

cosecant

In a right triangle with acute angle theta, the ratio comparing the length of the hypotenuse to the side opposite of theta. It is the reciprocal of the sine ratio.

cosine

In a right triangle with acute angle theta, the ratio comparing the length of the side adjacent to theta and the hypotenuse.

cotangent

In a right triangle with acute angle theta, the ratio comparing the length of the side adjacent to theta and the side opposite theta. It is the reciprocal of the sine ratio. It is the reciprocal of the tangent ratio.

sine

In a right triangle with acute angle theta, the ratio comparing the length of the side opposite theta and the hypotenuse.

tangent

In a right triangle with acute angle theta, the ratio comparing the length of the side opposite theta and the side adjacent to theta.

standard position

In the coordinate plane, an angle positioned so that its vertex is at the origin and its initial side is along the positive x-axis.

trigonometric ratios

Ratios that are formed using the side measures of a right triangle and a reference angle theta.

reference angle

The acute angle formed by the terminal side of an angle in standard position and the x-axis.

angle of elevation

The angle formed by a horizontal line and an observer's line of sight to an object above.

angle of depression

The angle formed by a horizontal line and an observer's line of sight to an object below.

terminal side

The final position of a ray after rotation when forming an angle.

damped trigonometric function

The function formed when a sinusoidal function y = sin bx or y = cos bx is multiplied by another function y = f(x). A function of the form y = f(x) sin bx or y = f(x) cos bx.

inverse trigonometric function

The inverse sine of x or sin^-1 x, the inverse cosine of x or cos^-1 x, and the inverse tangent of x or tan^-1 x.

linear speed

The rate at which an object moves along a circular path.

angular speed

The rate at which the object rotates about a fixed point.

initial side

The starting position of a ray when forming an angle.

reciprocal function

Trigonometric functions that are reciprocals of each other.

trigonometric functions

sin, cos, tan, csc, sec, cot

Law of Sines

sinA/a=sinB/b=sinC/c