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Complete Assignment 8, including Class Notes, text problems and Web-based problems as specified on the Assts page.When you have completed the entire assignment run the Query program. Submit SEND files from Query and q_a_.

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assignment #008

008. Identities

Precalculus II

11-12-2008

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20:31:20

Previous Assignments: Be sure you have completed Assignments 6 and 7 as instructed under the Assts link on the homepage at 164.106.222.236 and submitted the result of the Query and q_a_ from that Assignment. Note that Assignment 7 consists of a test covering Assignments 1-5.

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RESPONSE -->

ok

self critique assessment: 3

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20:44:12

`q001. Note that there are four questions in this Assignment.

In general the sine and cosine functions and tangent function are defined for a circle of radius r centered at the origin. At angular position theta we have sin(theta) = y / r, cos(theta) = x / r and tan(theta) = y / x. Using the Pythagorean Theorem show that sin^2(theta) + cos^2(theta) = 1.

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RESPONSE -->

sin=y/1+cos=x/1=1

sqrt(tan)=1

y/x=tan

a^2+b^2=c^2

confidence assessment: 2

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20:44:44

The Pythagorean Theorem applies to any point (x,y) on the unit circle, where we can construct a right triangle with horizontal and vertical legs x and y and hypotenuse equal to the radius r of the circle. Thus by the Pythagorean Theorem we have x^2 + y^2 = r^2.

Now since sin(theta) = y/r and cos(theta) = x/r, we have

sin^2(theta) + cos^2(theta) = (y/r)^2 + (x/r)^2 =

y^2/r^2 + x^2/r^2 =

(y^2 + x^2) / r^2 =

r^2 / r^2 = 1.

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RESPONSE -->

ok

self critique assessment: 2

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20:47:22

`q002. Using the fact that sin^2(theta) + cos^2(theta) = 1, prove that tan^2(theta) + 1 = sec^2(theta).

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RESPONSE -->

y/r+x/r=1

This isn't so.

(y/r)^2+(x/r)^2 =1, but

y/r+x/r is not equal to 1.

y/x+1=r/x

This does not follow from the your equation y/r+x/r=1. You need to show the steps of your algebra so I can tell where you went wrong.

confidence assessment: 2

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20:48:22

Starting with tan^2(theta) + 1 = sec^2(theta) we first rewrite everything in terms of sines and cosines. We know that tan(theta) = sin(theta)/cos(theta) and sec(theta) = 1 / cos(theta). So we have

sin^2(theta)/cos^2(theta) + 1 = 1 / cos^2(theta).

If we now simplify the equation, multiplying both sides by the common denominator cos^2(theta), we get

sin^2(theta)/cos^2(theta) * cos^2(theta)+ 1 * cos^2(theta)= 1 / cos^2(theta) * cos^2(theta).

We easily simplify this to get

sin^2(theta) + cos^2(theta) = 1,

which is thus seen to be equivalent to the original equation tan^2(theta) + 1 = sec^2(theta).

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RESPONSE -->

ok r = 1 because of the radius of the unit circle

self critique assessment: 2

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20:52:51

`q003. Prove that csc^2(theta) - cot^2(theta) = 1.

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RESPONSE -->

r/x-x/y=1

(r/x)^2-(x/y)^2=1, but

r/x-x/y is not equal to 1.

Therefore 1+x/y=r/x

r=1 on unit circle

confidence assessment: 2

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20:53:30

Rewriting in terms of sines and cosines we get

1 / sin^2(theta) - cos^2(theta)/sin^2(theta) = 1.

We now multiply through by the common denominator sin^2(theta) to get

1 / sin^2(theta) * sin^2(theta) - cos^2(theta)/sin^2(theta) * sin^2(theta) = 1 * sin^2(theta), or

1 - cos^2(theta) = sin^2(theta).

This is easily rearranged to give us sin^2(theta) + cos^2(theta) = 1, which we know to be true. The original equation is thus equivalent to this true equation, and is therefore true.

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RESPONSE -->

ok used x and ys instead of sin and cos

self critique assessment: 2

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20:55:28

`q004. Prove that sec^2(theta) * csc^2(theta) - csc^2(theta) = sec^2(theta).

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RESPONSE -->

sec*csc-csc=sec

csc subtracts to leave result

confidence assessment: 2

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20:56:28

Rewriting in terms of sines and cosines we get

1/cos^2(theta) * 1 / sin^2(theta) - 1/sin^2(theta) = 1/cos^2(theta).

We now multiply through by the common denominator sin^2(theta)* cos^2(theta) to get

sin^2(theta)* cos^2(theta) * 1/cos^2(theta) * 1 / sin^2(theta) - sin^2(theta)* cos^2(theta) * 1/sin^2(theta) = sin^2(theta)* cos^2(theta) * 1/cos^2(theta).

Simplifying we get

1 - cos^2(theta) = sin^2(theta), which we rearrange to get

sin^2(theta) + cos^2(theta) = 1.

Note that there are other strategies for proving identities, which you will see in your text.

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RESPONSE -->

ok misread problem as csc -csc

self critique assessment: 2

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You aren't quite getting the expressions right, and your are also making some algebra errors.

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