Trigonometry DAT quantitative reasoning

Therefore, the inverse sin function of x is defined only for x such that

-1<x<1

tan(x) is undefined in these examples

-630, -450, -270, -90, 90, 270, 450, 630 radians: (-7(pi)/2), (-5(pi)/2), (-3(pi)/2), (-(pi)/2), ((pi)/2), (3(pi)/2), (5(pi)/2), (7(pi)/2)

for example, for arcsin(1/2), to be able to assign a unique value to this, the requirement that ________ be included

-90<y<90

By the inverse tangent function of x, which is denoted by acrtan(x) or tan^-1(x), it is meant that angle y is such that

-90<y<90 and tan(y)=x

So in the interval -90<x<0, tan x is increasing. when x=____, tan x=0.

0

When 0 is less than or equal to x which is less than or equal to 90 degrees, sin x increases from 0 (when x=_____ degrees) to 1 (where x= ____)

0 90

When ____<x<______, tan x continues to inc from positive values that have a small absolute value for values of x that are near but greater than 0 degrees to positive that have a large absolute value for values of x that are close to but less than 90 degrees.

0<x<90

If a central angle in a circle intercepts an arc equal in length to 1 radius of a circle, that central angle has a radian measure of

1

The graph of cos(x) begins at what number?

1

The cos graph begins at 1 when x=____ and in the interval 0<x<90, it decreases from __ to ____.

1 1 0

how many degrees are in 5(pi)/6 radians?

150

The graph of tan repeats every how many degrees? (this is different from sin and cos(

180

when 180< x< 270, sin x decreases from 0 (when x=____ degrees) to -1 (where x= _____ degrees)

180 270

what is the number of radians in the circumference of any circle?

2(pi)

What is the distance between the top of one peak to another?

2(pi) or 360 degrees

when 270< x< 360, sin increases from -1 (where x=____ degrees) to 0 (where x=___ degrees)

270 360

When 270<x<360, cos(x) increases from 0 when x=___ to 1 when x=_____. Then for 360<x<720, the graph repeats itself again with one more _______

270 360 complete cycle.

because 2pi is the smallest possible positive value for R which sin (x+R)=sin(x) for all x, ____ is the period of sin x

2pi

2pi radians equals how many degrees?

360

how many degrees are in a full circle?

360

The graph of y=sin(x) repeats itself every interval of length _____

360 degrees (or 2pi radians)

the graph of y=cos(x) repeats itself every

360 degrees or 2(pi) radians

what is the conversion factor of radians to degrees

360 degrees/2(pi) radians

1 radian is equal to

360/2(pi) or 180/pi degrees this is approximately 57.3 degrees

sin x increases from 0 to 1 (in the interval ________) and decreases from 1 to 0 (in the interval ____) then continues to decrease from 0 to -1 (in the interval ____) and then increase from -1 to 0 (in the interval ____0

360<x<450 450<x<540 540<x<630 630<x<720

when 90<x<180 (all less than or equal to), sin x decreases from 0 (when x= ______ degrees) to -1 (when x= ______ degrees)

90 180

When 90<x<180, cos(x) continues its decrease from 0 when x=____ to -1 when x=____. When 180<x<270, cos(x) inc from -1 when x=____ to 0 when x=____.

90 180 180 270

the graph of y=tan(x) in the interval -90<x<90 is repeated in the interval ____________

90<x<270

what abbreviation helps you remember all the sine, cosine, and tangent things?

SOHCAHTOA

which trig functions would be undefined if x=0 and which trig functions would be undefined if y=0?

TAN and SEC COT and CSC

what is a degree

a unit of measure for describing angles

what is the cosine of an angle?

adjacent/hypotenuse

formula for cot?

adjacent/opposite

Notice that for the function f defined by f9x0= sin(x), sin(x+2pi)=sin(x) for

all x values

a radian is another way to describe an

angle

why are values -90 and 90 not possible values of the inverse tan function

because tan 90 and tan -90 are both undefined

Why does tan repeat every 180 degrees?

because tan(x) is positive for 0<x<90 and tan x is negative for 90<x<180, positive again 180<x<270 and negative again 270<x<360

Right after 360 degrees, sin x will

begin another cycle.

what are the abbreviations for these cotangent secant cosecant

cot sec csc

1/sin=

csc (cosecant)

A function is a set of instructions that associates each number of a set X (which is called the ____) with a number in the set Y (which is called _____)

domain range

What is the inverse tangent function defined for x?

for all real x

the reason that for a given x the value of arctan(x) is defined ro be in the interval -90<y<90 is that

for any given real number x, there are infinitely many values of y such that tan(y)=x and a function must associate exactly one number with each number in the domain

formula for sec?

hypotenuse/ adjacent

formula for csc?

hypotenuse/opposite

example, arctan(-1)= tan^-1(-1)= -45 degrees because -45 is

in the interval -90<x<90 and tan(-45)=-1

What is the reason for a given x, the value of arcsin(x) is defined to be in the interval -90<y<90 is that for a given x, there are

infinitely many y such that sin y=x, and a function must associate exactly one number with each number in the domain.

if a central angle of a circle intercepts an arc of a circle with length (l) then the number of radians in the central angle tha contains this arc is

l/r

while it is also true that sin(x+4pi)=sin(x), is 4pi the smalllest possible positive value for R such that sin (x+R)= sin x for all x

no

Tan(x) is undefined at every _____________

off multiple of 90 degrees or pi/2.

what is the tangent?

opposite/ adjacent

Let R be a positive constant. If f is a function such that f(x)= f(x+R) for all x, and R is the smallest positive number for which this is true, then the function is said to be

periodic with period R

In the tan graph, when x approaches ____ through values of x that are greater than the blank, tan x approaches negative infinity. if it is lesser then it approaches positive infinity.

pi/2

look up the unit circle hand trick

please

The x-axis of the graph of tan will be labeled with ______, unlike the graphs of sin and cos, which are labeled with _____

radians degrees

the trig identities are really just

restatements of the pythagorean theorem

Very important. for arcsin the interval is ___ and for arctan is ___

sin is -90 is less than OR EQUAL TO y and less than OR EQUAL TO 90 tan is -90<y<90

By the inverse sin function of x, which is denoted by arcsin x, or sin^-1(x), it is meant that the angle y is such that ___ and ____

sin y= x -90<y<90

Example, arcsin(1/2)=sin^-1(1/2)=30 degrees because

sin(30)=1/2 and 30 degrees is in the interval =90<y<90

because sin(x) repeats itself every 360, sin(30) is equal to

sin(390) ' both of these equal 1/2

to remember which goes with which (sin and cosine) vs (secant and cosecant)

the co needs to go with a not co so basically secant with COsine sine with Cosecant

what does f refer to?

the entire process of using the instructions to associate numbers with other numbers

what does f(x) refer to?

the exact number that the function f associated with the number x.

what is the sine of an angle?

the length of opposite the angle divided by the hypotenuse

what is the hypotenuse?

the side opposite the right angle

what forms the central angle?

two radii.

When 90<x<0, tan is negative, and tan increases from negative numbers with large/small absolute values whe x is near but less than 0.

very large

so even though for any x, sin x does have the same value at x+4pi that is has at x, we would/would not say that the period of f(x)sin x is 4pi/

would not because the period of sin x is 2pi

because -1<y<1, the inverse sin function cannot be defined for values such that

x<-1 or x>1