trig review3

8/10/2019 Trig Review3

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Everything youalways wanted toknow abouttrig*

Explained byJohn Baber, A.B.,Cd.E.

*but were afraid to ask.


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Any figure drawn on a piece of paper with straight edges can be split up into right triangles. Be-cause of this, studying squares, rectangles, rhombi, trapezoids, and all other regular and irregular polygons,can be seen as studying right triangles only. This is why there is a subject in school called trigonometry(measurement of triangles).

To study a right triangle fully, we have six functions defined to relate its sides and angles.

sin = opposite

hypoteneuse

cos = adjacent

hypoteneuse

tan = opposite

adjacent

csc =hypoteneuse

opposite

sec =hypoteneuse

adjacent

cot =adjacent

opposite

Really, sin and cos contain all of the information, the others are just shorthand.

tan = sin

cos csc =

1

sin sec =

1

cos cot =

cos

sin =

1

tan

You can memorize which is which by the old stand-by SOH-CAH-TOA*. i.e.

S

ine = O

pp/H

yp C

osine = A

dj/H

yp T

angent = O

pp/A

djThe fact that s = 1

candc = 1

s, tan, and cot, youll just have to get used to.

Since these were defined with being a non-right angle in a right triangle, they only make sense when0 < <

2. The only thing you ever do with degrees is translate them into radians: ( = 180 30 =

180

6 =

6etc.). So, graphs of sin and cos would look like

* Or you can use S

illy O

ld H

arry C

arried A

H

orse T

o O

ur A

partment if it bothers you that Sohcahtoais meaningless aside from its similarity to Krakatoa. . .


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Even though triangles with = 0 or = 2

dont make sense, by the graphs, we can tell what the valuesshould be. Heres an easy way to memorize all of the important values for sin, cos, and tan. Notice thepattern in

0 /6 /4 /3 /2 0 30 45 60 90

sin 0

/4 1

/4 2

/4 3

/4 4

/4

cos

4/4

3/4

2/4

1/4

0/4tan

0/4

1/3

2/2

3/1

4/0

So

cos 4

=

2

4 =

1

2= 1

2 =

2

2

To generalize these definitions so that can be any number, define sin and cos with a circle

Given an angle , find the point corresponding in the unit circle the x-coordinate will be cos andthey -coordinate will be sin . This is very nice for solving problems like

sin =3

2


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All solutions to the problem are

, + 2, + 4, . . .

2, 4, . . ., + 2, + 4, . . .

2, 4, . . .which is always shortened to

+ 2k and + 2k, k Z

In our case, we can examine the same triangle moved to the Iquadrant and figure out what is.

Looking at the table, see that is 3

, so = += + 3

= 43

and = 2 = 2 3

= 53

. So

all solutions would be these two angles +2k, k Z. A calculator, when asked for sin1 32 only tellsyou one of these angles, namely

3.

* means member of Z means the integers, i.e. . . . ,2,1, 0, 1, 2, . . .

So k Z means k in the set of integers i.e. k= . . . ,2,1, 0, 1, 2, . . .


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With the circular definition of sin and cos, we can have the graphs were more accustomed to

Notice that sin1 and cos1 would have to be (by flipping about the 45-degree line)

but these wouldnt be functions (since theyd fail the vertical line test), so a decision had to be made.


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This is why sin1 and cos1 look like

These are the functions your calculator uses, which explains why it misses of the answers to a simplequestion like:

Given

sin =3

2 ,

what is ?


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Now, how do we memorize all of the trigonometric identities were expected to know? First memorizethe pythagorean identity

sin2 + cos2 = 1

Immediately, you can see that this means

sin2

= 1 cos2

cos2 = 1 sin2

Derive the two similar identities by dividing by sin2 on both sides of the pythagorean identity

1 + cot2 = csc2

and dividing by cos2 on both sides of the pythagorean identity

tan2 + 1 = sec2

Now memorize the angle addition formulae and you dont have to memorize anything else. The rest can

be derived. These are easier to understand when you say them out loud than when you read them on paper.So, say out loud Sine sucks

sin(x+y) = sin x cos y+ cos x sin y (s c c s = s

uc

c

s

= sucks)

Cosine kicks

cos(x+y) = cos x cos y sin x sin y (c c s s = c

ic

s

s

= kicks)

make sure to memorize which one has a minus sign in the middle and which doesnt.Now you can derive every other identity anybody has ever expected of you

sin(2x) = sin(x+x) = sin x cos x+ cos x sin x= 2 sin x cos x

cos(2x) = cos(x+x) = cos x cos x sin x sin x= cos2 x sin2 x= cos2 x sin2 x= cos2 x (1 cos2 x) = cos2 x 1 + cos2 x= 2 cos2 x 1= (1 sin2 x) sin2 x= 1 2sin2 x

From the last two, you can also see that

cos2x= 1 2sin2 x sin2 x= 1 cos2x2

= 2 cos2 x 1 cos2 x= 1 + cos 2x2

Work all of the above derivations out on a piece of paper yourself. Once youve done it once, you canthelp but be able to do them in the future.

trig review3

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