SOLUTION: Use the sum or difference to find the exact value of tan105 degrees. please help? Thank u.

Question 421010: Use the sum or difference to find the exact value of tan105 degrees.
please help? Thank u.
Answer by jsmallt9(3759) (Show Source): You can put this solution on YOUR website!
Whenever the phrase "exact value" is used in referring to the value of a trig function, it means that you must use special angles because exact values are only available for the special angles.

The special angles are all the integer multiples of 30 and 45 degrees. The exact value for any of these angles can be found.

In your problem, 105 degrees is not a multiple of 30 or 45 so it is not one of the special angles. But, as the problem mentions, it is possible to express 105 as the sum or difference of special angles. There are many possible sums or differences of special angles that result in 105. One of them is:
135 - 30 = 105
135 is special because it is 3*45 and 30 special and this difference of special angles results in 105. So we will use this difference and the formula for the tan of a difference:

Using 135 for A and 30 for B in this formula we getL

The left side simplifies to tan(105). On the right side we find the exact values for the tan of the special angles:

To eliminate the fractions within the fraction on the right side, I am going to multiply the numerator and denominator by 3:

which simplifies to:

And if we want a rational denominator we need to multiply the numerator and denominator by :

which simplifies as follows:

This is the simplified, exact value for tan(105).
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