Ch. 5: Trigonometric Functions

Positive angles

Generated by counter-clockwise rotation.

Pythagorean Identities

(cos theta)^2 + (sin theta)^2 = 1 1 + (tan theta)^2 = (sec theta)^2 (cot theta)^2 + 1 = (csc theta)^2

Quadrant II

x < 0, y > 0 Only sin and csc are positive.

Quadrant IV

x > 0, y < 0 Only cos and sec are positive.

Quadrant I

x > 0, y > 0 All trig functions are positive.

45 - 45 - 90 triangle (isosceles right triangle)

a / c = rad 2 / 2 sin 45 = sin (pi / 4) = rad 2 / 2 cos 45 = cos (pi / 4) = rad 2 / 2 tan 45 = sin / cos = 1

A pizza has a radius of 1 unit. So its circumference is 2(pi).

- Cut the pizza in 1/2, we get C = pi. - Cut the pizza in 1/4, we get C = pi/2. - Into 1/8 pieces, we get pi/4. - Into 1/12 pieces, we get pi/6.

Converting from degree to radian measure or vice versa

1) Each circle has 360 degrees. 2) The circumference of a circle is 2(pi)r. 3) The angle with a line through it = s/r = circumference/r = 2(pi)r/r = 2(pi). 4)360 = 2(pi) or 180 = pi.

Using reference angles to evaluate trig functions

1) Find theta prime 2) Identify quadrant 3) Find value

Evaluating trig functions

1) Identify quadrant 2) Identify coordinates 3) Find missing length 4) Find values

Ray

A part of a line that has only one endpoint and extends forever in the opposite direction.

Reference angle

A positive acute angle theta prime formed by the terminal side of theta and the x-axis. 0 < theta prime < 90 Used to help make finding trig values easier. Terminal side to closest x-axis.

Finding a positive and negative coterminal angle

Add 360n. Subtract 360n.

Cofunctions

Any pair of trigonometric functions of f and g for with f(theta) = g(90 degrees - theta) and g(theta) = f(90 degrees - theta). When prefix is "co"

Angle

Formed by two rays that have a common endpoint.

Domain and range

Domain of sin: (negative infinity, positive infinity) Range of sin: [-1, 1] Domain of cos : (negative infinity, positive infinity) Range of cos: [-1, 1]

Odd and Even functions

Even function: f(-x) = f(x) and Odd function: f(-x) = -f(x) Cos theta and sec theta are even functions. Sin theta, csc theta, tan theta, cot theta, and cot theta are odd functions.

Negative angles

Generated by clockwise rotation.

30 - 60 - 90 triangle

Half of equilateral triangle so a = 0.5(c) 30 degree angle = pi / 6 60 degree angle = pi / 3 b / c = rad 3 / 2 sin 60 = sin (pi / 3) = rad 3 / 2 sin 30 = sin (pi / 6) = 1 / 2

Complements and cofunctions

If we know alpha, then beta = 90 - alpha. Note: sin alpha = y / r and cos beta = y / r So to find complements, use the cofunction and then subtract theta from 90 degrees or pi / 2

Acute angle

Measure less than 90 degrees.

Right angle

Measures 90 degrees.

Obtuse angle

Measures greater than 90 degrees.

Unit circle

Radius = 1 unit y = sin x = cos

Angle of elevation

The angle formed by a horizontal line and the line of sight to an object that is above the horizontal line.

Angle of depression

The angle formed by the horizontal line and the line of sight to an object that is below the horizontal line.

The vertex of the angle

The common endpoint of an angle's initial side and terminal side.

The radian measure of central angles (theta)

The length ("s") of the intercepted arc divided by the circle's radius ("r"). s/r

Hypotenuse

The side opposite the right angle. Always the longest side.

Two positive angles are complements if

Their sum is 90 degrees or pi / 2

Periodic functions

This means that if the function values repeat after a certain amount of time/period. If you notice, sin and cos repeat after 1 revolution of the circle, which equals 360 degrees or 2pi. This is the same for cos and sec. On the other hand, tan and cot have a period of 180 degrees or pi.

Coterminal angles

Two angles with the same initial and terminal sides but possibly different rotations.

Standard position

Vertex is at the origin and the initial side is the positive x-axis.

Cosine function

cos theta = length of side adjacent to angle theta / length of hypotenuse = adjacent / hypo

Cotangent function

cot theta = length of side adjacent to angle theta / length of side opposite to angle theta = adjacent / opposite

Fundamental Identities

csc theta = 1 / sin theta sec theta = 1 / cos theta cot theta = 1 / tan theta

Cosecant function

csc theta = length of hypotenuse / length of side opposite to angle theta = hypo / opposite

Theta

s / r or s = r(theta)

Secant function

sec theta = length of hypotenuse / length of side adjacent to angle theta = hypo / adjacent

Sine function

sin theta = length of side opposite to angle theta / length of hypotenuse = opposite / hypo

Tangent function

tan theta = length of side opposite to angle theta/ length of side adjacent to angle theta = opposite / adjacent

Quotient identities

tan theta = sin theta / cos theta cot theta = cos theta / sin theta

Linear speed

v = s / t v: velocity = distance / t s: arc length

Angular speed

w = theta / t

Quadrant III

x < 0, y < 0 Only tan and cot are positive.