SOLUTION: What is the exact value of tan(15 degrees)? The correct answer is one of the following. Which one? A) 1- square root of 3 B) 2- 2(square root of 3) C) 2+ square root of

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Question 1198656: What is the exact value of tan(15 degrees)?
The correct answer is one of the following. Which one?
A) 1- square root of 3
B) 2- 2(square root of 3)
C) 2+ square root of 3
D) 2+ 2(square root of 3)
E) 2- square root of 3
Found 4 solutions by Alan3354, Theo, MathTherapy, math_tutor2020:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
What is the exact value of tan(15 degrees)?
The correct answer is one of the following. Which one?
A) 1- square root of 3
B) 2- 2(square root of 3)
C) 2+ square root of 3
D) 2+ 2(square root of 3)
E) 2- square root of 3
======================================
tan(15) is positive.
A is negative
B is negative
----
tan(15) < 1
C is > 1
D is > 1
----
That leaves E.
Use a calculator to confirm it.
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
selection E is your solution.

the identity formula used to find that is tan(x/2) = (1 - cos(x) / sin(x)

when x/2 = 15 degrees, x = 30
cos(x) = sqrt(3)/2
sin(x) = 1/2

(1 - cos(x) / sin(x) = (1 - sqrt(3)/3) / (1/2) = 2 - sqrt(3).
that's selection E.

you can confirm using your caalculator.
tan(15) = .2679491924
2 - sqrt(3) = the same.
(1 - cos(30))/sin(30) = the same.

here's a reference.
https://www.cuemath.com/half-angle-formula/
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

What is the exact value of tan(15 degrees)?
The correct answer is one of the following. Which one?
A) 1- square root of 3
B) 2- 2(square root of 3)
C) 2+ square root of 3
D) 2+ 2(square root of 3)
E) 2- square root of 3
You can use a few identities: 
1) Difference of TWO angles:   <==== Too COMPLEX and TIME-CONSUMING

2) Difference of TWO angles:   <==== Too COMPLEX and TIME-CONSUMING

3) HALF-ANGLE:   <==== EASIER to manipulate
OR
4) HALF-ANGLE:   <==== EASIEST to manipulate

There're other COMBINATIONS but they are too diffucult, time-consumig, and don't make any sense when there're much easier ways to get the answer!

Using the 4th, we get: HALF-ANGLE:  
                                                                                                       
                                                          CHOICE E)
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I'll use a geometric approach, and a bit of algebra.

What we can do is form this rectangle here:

Note the 15+45+30 = 90 in the top right corner.

The three triangles at the bottom are your standard template 30-60-90 triangles, and a 45-45-90 triangle.
We'll focus on the triangle up top. It has an angle of 15 degrees.
The tangent of this angle is the opposite over adjacent.





Multiply top and bottom by some variable, which I'll call A.

Plug in so that the denominator is rationalized.

Expand the numerator. Use the difference of squares rule in the denominator.

The square root is canceled out in the denominator.







Answer: Choice E
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