MAC1147 Exam 3 Concepts

Phase Shift

A horizontal translation of a periodic function; Calculate as c/b.

Bearings

Based on compass Read traditionally left to right

Function vs. One-to-One

Function - each x has only one y, passes vertical line test one to one- each y has only one x, passes horizontal line test

The period of y= tan x is

Pi

arc =

^-1x

One period of a sine or cosine function is one of the sine or cosine curve.

cycle

a^3 + b^3/ a^3 - b^3

(a + b)(a^2 - ab + b^2) (a - b)(a^2 + ab + b^2)

log or ln cannot be = to

0

Guidelines for Verifying Trigonometric Identities

1. Start with one side. It's usually easier to start with the more complicated side. 2. Use known identities. Bring fractional expressions to a common denominator, factor, and use the fun- damental identities to simplify expressions. 3. Convert to sines and cosines. Sometimes it is help- ful to rewrite all functions in terms of sines and cosines. Sometimes, it is practical to work with each side sepa- rately to obtain one common form equivalent to both sides.

Period

2pi/b of y = a sin(bx + c)

The of a sine or cosine curve represents half the distance between the maximum and minimum values of the function.

Amplitude

What do you do if they don't want it in fractional form?

Multiply by the conjugate. Might have to separate into smaller components.

Does sin 60 degrees/sin 30 degrees = sin 2 degrees?

No

Does tan[(5 degrees)^2] = tan^2(5 degrees)

No, tan(25 degrees) does not = (tan 5 degrees)(tan 5 degrees)

Dividing by 0 (1/0) is

Not possible; undefined.

Sec is even, odd, or neither?

Odd

A function f is when f(-t) = -f(t) and when f(-t) = f(t).

Odd; Even

To sketch the graph of a secant or cosecant function, first make a sketch of its Function.

Reciprocal

The acute angle formed by the terminal side of an angle theta in standard position and the horizontal axis is the angle of theta and is denoted by theta' .

Reference

How to Take the Inverse of A Trigonometric Function

Set the number given to y, for sin, x for cos, or the y/x for tangent and solve for place on unit circle equal to that number. Have to remember domain and range restrictions.

Two separate triangles together?

Solve each and then add together to equal what you know

Periodic Function

To solve problems, add period of that function to angle to find smallest coterminal angle and then solve from there

True or False: You can obtain the graph of y = csc x on a calculator by graphing the reciprocal of y = sin x.

True. For a given value of x, the y-coordinate of csc x is the reciprocal of the y-coordinate of sin x.

True or False: You can obtain the graph of y = sec x on a calculator by graphing a translation of the reciprocal of y = sin x.

True. y = sec x is equal to y=1/cos x and if the reciprocal of y=sin x is translated pi/2 units to the left, then 1/sin(x+pi/2) = 1/cosx = sec x.

In theoretic situations:

Use numbers to test

How fast is an object moving?

Use the side that has already been given to you with how fast it moves

Inverse Trigonometric Function

Used to find a missing angle measure. (sin⁻¹, cos⁻¹, tan⁻¹)

The graphs of tangent, cotangent, secant, and cosecant functions have asymptotes.

Vertical

Pythagorean Identities

You can additionally factor these out

Amplitude

absolute value of a in y = a sin(bx+c)

Cofunctions of angles are equal.

complementary

hypotenuse/opposite

cosecant (csc)

adjacent/hypotenuse

cosine (cos)

adjacent/opposite

cotangent (cot)

An angle of represents the angle from the horizontal upward to an object, whereas an angle of represents the angle from the horizontal downward to an object.

elevation; depression

Shadow on a Triangle Problem

is the bottom of the triangle

Relative to the acute angle theta , the three sides of a right triangle are the side, the side, and the .

opposite; adjacent; hypotenuse

For the function y= a sin(bx-c), c/b represents the of one cycle of the graph of the function.

phase shift

Cofunction Identities

pi/2 = 90 degrees

When triangle are similar, their corresponding sides are

proportional

hypotenuse/adjacent

secant (sec)

Basic Trigonometric Functions

sin = y, cos = x, tan = y/x, csc = 1/y, sec = 1/x, cot = x=y

opposite/hypotenuse

sine (sin)

opposite/adjacent

tangent (tan)

For the function y = d + a cos (bx - c), d represents a of the basic curve.

vertical translation

sqrt of x^2 =

x

Example of Vertical Stretch Arc Sin Function

y = 2 arc sin x

Even-Odd Properties of Trigonometric Functions

cos and sec are even (cos (-t) = cos (t)), all other functions are odd (tan(-t) = -tan (t)).

Trig Inverse Properties

the simplified inside has to be in domain of outside trig function

Conditional Equation vs. An Identity

A conditional equation an equation that is true for only some of the values in its domain. ex. tanx=1 An identity is an equation that is true for all real values in its domain. ex. sin2x=1−cos2x

Angles Cut by Transversals

Equal Each Other

To Solve A Right Triangle Means

Find the Missing Lengths of Its Sides and the Measurement of its Angles a^2 + b^2 = c^2 A + B = 90 degrees

The tangent, cotangent, and cosecant functions are , so the graphs of these functions have symmetry with respect to the .

Odd; origin

A function f is when there exists a positive real number c such that f(t+c)=f(t) for all t in the domain of f.

Periodic

Period of tangent & cotangent functions

Pi/b