SOLUTION: find the exact value of tan 15degrees without using tables or a calculator

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Question 207450: find the exact value of tan 15degrees without using tables or a calculator
Found 2 solutions by Edwin McCravy, jsmallt9:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Find the exact value of tan(15°) without using tables or a calculator

%2215%B0%22 is not a special angle, however %2245%B0%22 and %2230%B0%22
are.  And %2215%B0%22=%2245%B0%22-%2230%B0%22, so we can use this formula:



where alpha=%2245%B0%22 and beta=%2230%B0%22:



Multiply top and bottom by 3



Rationalize the denominator:

=

.

Check by using your calculator.  

tan%28%2215%B0%22%29=+.2679491924

Then find the decimal value for

2-sqrt%283%29+=+.267941924

So it checks.

So the answer is 

tan%28%2215%B0%22%29=2-sqrt%283%29

Edwin

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The only "exact values" for Trigonometric functions which can be found are for 0, 30, 45, 60, 90 degress (and for angles whose reference angles are these). So the only way to find the excat value of tan(15) is to express the tan(15) in terms of these other angles.
Since 15 = 1/2 of 30 we could use the 1/2 angle formula:
tan%281%2F2%29x+=+%281+-+cos%28x%29%29%2F%28sin%28x%29%29
where x = 30:
tan%281%2F2%2930+=+%281+-+cos%2830%29%29%2F%28sin%2830%29%29
Since cos%2830%29+=+sqrt%283%29%2F2 and sin%2830%29+=+1%2F2:
tan%2815%29+=+%28%281+-+sqrt%283%29%2F2%29%29%2F%281%2F2%29
Change the division by 1/2 to mutiplying by 2/1:
tan%2815%29+=+%281+-+sqrt%283%29%29+%2A+%282%2F1%29
tan%2815%29+=+2+-+sqrt%283%29

Another way would be to use the fact that 15 = 45 - 30 and use the difference formula
tan%28a-b%29+=+%28tan%28a%29+-+tan%28b%29%29%2F%281%2Btan%28a%29+tan%28b%29%29
with a = 45 and b = 30:
tan%2845-30%29+=+%28tan%2845%29+-+tan%2830%29%29%2F%281+%2B+tan%2845%29+tan%2830%29%29
Since tan%2845%29+=+1 and tan%2830%29+=+sqrt%283%29%2F3:
tan%2815%29+=+%281+-+sqrt%283%29%2F3%29%2F%281+%2B+1+%2A+%28sqrt%283%29%2F3%29%29
Multiplying the top and bottom by 3:
tan%2815%29+=+%283+-+sqrt%283%29%29%2F%283+%2B+sqrt%283%29%29
Next we rationalize the denominator by mulitplying by its conjugate: %283+-+sqrt%283%29%29:
tan%2815%29+=+%289+-+6%2Asqrt%283%29+%2B+3%29%2F%289+-+3%29
tan%2815%29+=+%2812+-+6%2Asqrt%283%29%29%2F6
Factoring out 6 on top:
tan%2815%29+=+%286%282+-+sqrt%283%29%29%29%2F6
Canceling the common factor of 6:
tan%2815%29+=+2+-+sqrt%283%29