Expert: Scotto - 3/6/2009

Question
QUESTION: I can not figure this question out it says:

Use a double or half angle formula to find the exact value of tan Pi/12

ANSWER: The exact value is known of that tan(π/3).

Using the half angle formula on this will give you the tan(π/6).
Using the half angle formula on this will give you the tan(π/12).
tan(A/2) = sin(A)/(1 + cos(A)).

The tan(π/3) is √3/1 = √3.
The sin(π/3) is √3/2.
The cos(π/3) is 1/2.

Using half angle formula for sin() and cos()
can be used to find the sin(π/6) and cos(π/6).

Here are the half angle formulas:
It is know that sin(Θ/2) = ±√(1 - cos(Θ).
and that the cos(π/2) = ±√(1 + cos(Θ)).

With some manipulation, the tan(π/12) can be found.


---------- FOLLOW-UP ----------

QUESTION: I don't understand this at all from there where do i go about now with this u said with some manipulation i can find out what manipulation are u talking about

Answer
A² is A squared = A*A.
√A is the square root of A.

The equations to use are

tan(B/2) =  (1−cos B) / sin B .

Here, the angles are in degrees.  Note that π radians = 180 degrees.  We are trying to find tan(π/12),
which is 15 degrees.

If we look at an equilateral triangle we know all of the sides are 1 and the angles are all 60 degrees.  Have the triangle pointing up with a flat side on the bottom.  If we bisect the top angle and bring it down to the middle side, the 60 degree angle at the top is cut into a 30 degree angle.  This a 30-60-90 triangle.  The hypotenuse is 1 and the side on the bottom is 1/2.

We know that c² - b² = a².  We know that c² = 1, b² = 1/4, so that makes a² = ¾.  From here, we can say that a=√3/√4 = √3/2.  From here, we can say that since the angle at the top, which was 60, was cut in half by our vertical line and the angle in each triangle is 30.  So with this 30 degree angle, we have
Sin(30) = ½ and cos(30) = √3/2.

I will make a picture up and send it as well.

We will now use the equation and say tan(30/2) = (1-cos(30))/sin(30).
This is tan(15) = (1 - √3/2)/(1/2).

Remember in basic math, multiplication and division are done before subtraction and addition.

Also, the value in each set of parenthesis needs to be calculated before the overall division is done.

√3/2 is approximately 0.866.  When this is subtracted from 1, you should get 0.134.  Dividing this expression by ½ is the same as multiplying it by 2, so the answer is 0.268.

But to be exact, leave it in the form (2 - √3).