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003.

Precalculus II

10-26-2008

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assignment #005

005.

Precalculus II

10-26-2008

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20:09:23

Query problem 6.2.8 exact value of tan(195 deg)

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RESPONSE -->

.224

confidence assessment: 2

.224 is a calculator approximation, not an exact value. The calculator cannot give an exact value, which is an irrational number. It can only approximate.

tan(alpha+beta)= [tan(alpha)+tan(beta)]/[1-tan(alpha)tan(beta].

So tan(195 deg) = tan(150 deg + 45 deg) =
[ tan(150 deg) + tan(45 deg) ] / [ 1 - tan(150 deg) * tan(45 deg) ] =
[ -sqrt(3) / 3 + 1 ] / [ 1 - (-sqrt(3)) * 1 ] =
[1 - sqrt(3)/3] / (1 + sqrt(3)/3 )=
(1 - sqrt(3)/3)(1 - sqrt(3)/3) / [ (1+sqrt(3)/3)(1-sqrt(3)/3) ] =
(1 - 2 sqrt(3)/3 + 1/3) / (1 - 1/3) =
(4/3 - 2 sqrt(3)/3) / (2/3) =
2 - sqrt(3).

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20:09:36

what is the exact value of tan(195 deg)?

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RESPONSE -->

.224

confidence assessment: 2

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20:22:49

to get this result you used the exact values of two angles to get tan(15 deg). Which two angles were these, what were the exact values, and what formula did you use to get your result?

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RESPONSE -->

tan=13pi/12

3.4=tan 195 deg

confidence assessment: 2

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20:26:24

For which angles between 0 and 90 degrees do you know the exact value of the trigonometric functions? How many ways are there to combine angles you have listed to obtain 15 degrees?

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RESPONSE -->

5 trig functions

there are 24 ways to obtain 15 deg angles because when pi is divided by 12 this yields 15 deg.

confidence assessment: 2

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20:29:07

Query problem 6.2.20 cos(5 `pi / 12) cos(7`pi/12) - sin(5`pi/12)sin(7`pi/12)

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RESPONSE -->

.2588

-.2588

.9659

-.9659

These angles are inverses due to what correlation they have on the unit circle

confidence assessment: 2

Good student solution:

to begin solving this equation it is obvious that we need to simplify.
because each part to this equation can be simplified using the sum and
difference formulas. first we take cos(5pi/12) this can be written as
cos(75deg) we then know that

cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta).

so we can say

cos(45+30)=cos45cos30-sin45sin30.

(sq.rt.2)/2(sq.rt.3)/2- (sq.rt.2)/2(1/2)

which gives us

(sq.rt.6-sq.rt.2)/4.

we can then simplify the second term in the equation. cos7pi/12. which can
be written as cos(105deg). we can then say

cos(60+45)=cos60cos45-sin60sin45.

which is

(1/2)(sq.rt.2/2)-(sq.rt.3/2)(sq.rt.2/2)=(sq.rt.2-sq.rt.6)/4.

we can now solve for the third term in the equation which is sin(5pi/12) which can
be written as sin75deg. we can then say that

sin(45+30)=sin45cos30+cos45sin30. which is equal to

(sq.rt.2/2)(sq.rt.3/2)+(sq.rt.2/2)(1/2)=(sq.rt.6+sq.rt.2)/4.

we can now solve the fourth term in the equation which is sin7pi/12. which can be
written as sin 105 deg. we can then say

sin(60+45)=sin60cos45+cos60sin45=(sq.rt.3/2)(sq.rt.2/2)+(1/2)(sq.rt.2/2)
=(sq.rt.6+sq.rt.2/4).

we can now solve the equation. (sq.rt.6-sq.rt.2)/4 (sq.rt.2-sq.rt.6)/4
-(sq.rt.6+sq.rt.2)/4(sq.rt.6+sq.rt.2)/4=
(2sq.rt.12)-8/16- (2sq.rt.12)+8/16= 0 .

INSTRUCTOR CORRECTION: great until the last step, which should read

(2sq.rt.12)-8/16- [(2sq.rt.12)+8/16]= -1.

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20:31:17

what is the exact value of the expression?

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RESPONSE -->

the exact values are pos and neg .2588 and .9659

confidence assessment: 2

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20:31:28

How did you obtain your result?

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RESPONSE -->

by using a calculator

confidence assessment: 3

Your calculator rounds things off. The exact values are irrational numbers and cannot be expressed accurately by your calculator.

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20:34:06

Which sum or difference formula does this expression fit?

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RESPONSE -->

this formula fits the differance formula because there is a differance between the values that make the functions both pos and neg

confidence assessment: 2

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20:35:32

If this expression is the right-hand side of a sum or difference formula, what then is the left-hand side and what is its exact value?

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RESPONSE -->

the exact formula of difference is shown because the 5pi/12 is on the right side and the 7pi/12 is on the left hand side of the y axis

confidence assessment: 2

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20:39:46

Query problem 6.2.38 tan(2 `pi - `theta) = - tan(`theta)

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RESPONSE -->

theta=2pi

confidence assessment: 1

Good student solution:

to prove this identity we can say that tan(2pi-theta)=-tan theta. to figure
out this problem we use the sum and difference formula for

tan(alpha-beta)=tan 2pi-tan theta/1+tan2pi tan theta

we know that the value of tan of 2pi =0 so we can say

0-tan theta/1+(0)(tan theta=(- tan theta)/1

which is -tan theta so the identity is true.

INSTUCTOR COMMENT AND ALTERNATIVE SOLUTION: An alternative would be to use the periodicity and even/odd properties of sine and cosine. sin(-`theta) = - sin(`theta), cos(-`theta) = - cos(`theta) so tan (-`theta) = - tan(`theta); all functions have the same values at 2 `pi + `theta as at `theta, so the result follows.

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20:41:36

how do you establish the given identity?

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RESPONSE -->

by canceling the tan and neg values and x values and this yields the only thing left being 2pi

confidence assessment: 2

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20:51:43

Query problem 6.3.8 find exact values of sin & cos of 2`theta, and of `theta/2 if csc(`theta) = -`sqrt(5)

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RESPONSE -->

csc=y/1

-sqrt5=2.236

csc(x)=-2.236

x=-2.236

sin(2x)=1/y(2x)=-.4128

Cos(2x)=1/x(2x)=.9108

confidence assessment: 1

sin^2(`theta) + cos^2(`theta) = 1 so

cos^2(`theta) = 1 - sin^2(`theta) and

cos(`theta) = `sqrt[ 1 - sin^2(`theta) ] = `sqrt(1 - (`sqrt(5) / 5)^2 ) = `sqrt(1 - 1/5)^2 = `sqrt(4/5) = 2 `sqrt(5) / 5.

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20:52:39
what are the exact values of the given expressions?

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RESPONSE -->
the exact values are .9108, -.4128 of the sin and cos functions given
confidence assessment: 1

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20:53:53
Which formulas did you used to obtain these results?

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RESPONSE -->
the formula used derived the answers from used the graphs of the functions
confidence assessment: 1

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20:54:55
Explain how the identities sin(2`theta) = 2 sin(`theta) cos(`theta), cos(2`theta) = cos^2(`theta) - sin^2(`theta), sin(`alpha/2) = +- `sqrt( (1 - cos(`alpha) ) / 2), cos(`alpha) = +-`sqrt( (1 + cos(`alpha) ) / 2) are used to get your results.

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RESPONSE -->
these functions are the inverses of the sin cos and tan functions so instead of 1/x or 1/y they are y/1 or x/1
confidence assessment: 2

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20:58:52
Query problem 6.3.20 exact value of csc(7`pi/8)

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RESPONSE -->
7pi/8=315 deg
csc315=1/2
confidence assessment: 2

** sin(7`pi/8) = sin( 1/2 * 7`pi/4). You know the exact value of sin(7`pi/4). Use the half-angle formula. You get `sqrt(2`sqrt(2)+4) ). **

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20:59:14

what is the exact value the given expression?

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RESPONSE -->

2.7488

confidence assessment: 2

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21:00:20

Explain how you obtained your result from the half-angle formulas.

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RESPONSE -->

this result was derived from the division of the unit circle into eights and then form there the 5pi/6 was used and inserted into the csc function yielding the result

confidence assessment: 3

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21:06:05

query problems 6.3.42 tan(`theta/2) = csc(`theta) - cot(`theta)

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RESPONSE -->

x(x/2)=y/x

cot=sec(x/2)

confidence assessment: 1

The given formula in the text is

tan(`alpha/2)= (1-cos (`alpha)) / sin (`alpha) which is the same as

1 / sin(`alpha) - cos(`alpha) / sin(`alpha) =
csc(`alpha) - cot(`alpha).

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21:06:29

explain how you established the given identity

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RESPONSE -->

by using x and y to simplify

confidence assessment: 2

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21:09:49

Comm on any surprises or insights you experienced as a result of this assignment.

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RESPONSE -->

this assignment has really challenged me and made me realize that i need to study my half formulas

confidence assessment: 3

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assignment #005

005.

Precalculus II

10-26-2008

You should rework this assignment using the correct procedures and identities.

&#Please see my notes and submit a copy of this document with revisions and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end). &#

Your calculator rounds things off. The exact values are irrational numbers and cannot be expressed accurately by your calculator.

** sin(7`pi/8) = sin( 1/2 * 7`pi/4). You know the exact value of sin(7`pi/4). Use the half-angle formula. You get `sqrt(2`sqrt(2)+4) ). **

** The given formula in the text is tan(`alpha/2)= (1-cos (`alpha)) / sin (`alpha) which is the same as 1 / sin(`alpha) - cos(`alpha) / sin(`alpha) = csc(`alpha) - cot(`alpha). **

You should rework this assignment using the correct procedures and identities.

&#Please see my notes and submit a copy of this document with revisions and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end). &#

The given formula in the text is

tan(`alpha/2)= (1-cos (`alpha)) / sin (`alpha) which is the same as

1 / sin(`alpha) - cos(`alpha) / sin(`alpha) =
csc(`alpha) - cot(`alpha).