SOLUTION: Find sinA, cosA, and tanA of the following triangle. Line AC=2 Line AB=4 Angle C is a right triangle (ie. 90 degrees) Thanks!

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Question 88982: Find sinA, cosA, and tanA of the following triangle.
Line AC=2
Line AB=4
Angle C is a right triangle (ie. 90 degrees)
Thanks!
Found 2 solutions by Edwin McCravy, bucky:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Find sinA, cosA, and tanA of the following triangle.
Line AC=2
Line AB=4
Angle C is a right triangle (ie. 90 degrees)
Thanks!



Now we use the Pythagorean theorem to find the missing side BC:

AC² + BC² = AB²

 2² + BC² = 4²

  4 + BC² = 16

      BC² = 12
             __
       BC = Ö12
             ___
       BC = Ö4·3 
              _
       BC = 2Ö3

So now the triangle is:



sin%28A%29 = +%28OPPOSITE%29%2F%28HYPOTENUSE%29+ = %28BC%29%2F%28AB%29 = %282sqrt%283%29%29%2F4 = sqrt%283%29%2F2

cos%28A%29 = +%28ADJACENT%29%2F%28HYPOTENUSE%29+ = %28AC%29%2F%28AB%29 = 2%2F4 = 1%2F2

tan%28A%29 = +%28OPPOSITE%29%2F%28ADJACENT%29+ = %28BC%29%2F%28AC%29 = %282sqrt%283%29%29%2F2 = sqrt%283%29

Edwin

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
If you sketch triangle ABC as you described (with angle C the right angle) you can see that
side AC is one leg of the triangle and side AB is the hypotenuse. Side BC is the other leg
and is of unknown length.
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Use the Pythagorean theorem to find the length of side BC.
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%28BC%29%5E2+%2B+%28AC%29%5E2++=+%28AB%29%5E2
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Since the problem gives you the values for AC and AB you can substitute those values and
the Pythagorean equation becomes:
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%28BC%29%5E2+%2B+2%5E2+=+4%5E2
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Square out the known terms to get:
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%28BC%29%5E2+%2B+4+=+16
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Subtract 4 from both sides to get:
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%28BC%29%5E2+=+12
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Solve for BC by taking the square root of both sides to get:
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BC+=+sqrt%2812%29+=+sqrt%284%2A3%29+=+sqrt%284%29%2Asqrt%283%29+=+2%2Asqrt%283%29
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So now you know all three sides. The hypotenuse is 4, the side adjacent to angle A is 2,
and the side opposite to angle A is 2%2Asqrt%283%29.
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Now all you have to do is to use the definitions for sin A, cos A, and Tan A as follows:
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Sin%28A%29+=+opposite%2Fhypotenuse+=+%282%2Asqrt%283%29%29%2F4+=+sqrt%283%29%2F2
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Cos%28A%29+=+adjacent%2Fhypotenuse+=+2%2F4+=+1%2F2
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Tan%28A%29+=+opposite%2Fadjacent+=+%282%2Asqrt%283%29%29%2F2+=+sqrt%283%29
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As you gain a little experience you might recognize that the given triangle is a 30 - 60 - 90
triangle where 30 degrees is angle B, 60 degrees is angle A, and 90 degrees is angle C.
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Therefore, you can check these answers by entering 60 degrees on a scientific calculator.
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Then press the sin key to find that sin(60) = 0.866025403. This will be the same answer that
you will get by taking the square root of 3 and dividing it by 2.
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Similarly if you enter 60 degrees and press the cos key to find that cos(60) = 0.5
This equals the above answer that cos(A) = 1/2.
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Finally, if you enter 60 degrees and press the tan key to find that tan(60) = 1.732050808
and this is identical to the answer that tan(A) = square root of 3 that we got above.
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Hope this helps you to see how you can work this problem.
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