Time remaining:
Find the exact value of tan pi/12 + cos(3pi/2 - pi/12)

Calculus
Tutor: None Selected Time limit: 0 Hours

Please show all work. Thank you!

Jan 28th, 2015

So, we're going to need some formulas here. There are several formulas for the tangent of a half angle (I'm choosing that one because pi/12 is not an angle of a "special" triangle, but pi/6 is! pi/6 is equal to 30°. (pi radians equals 180°) and pi/12 is half of pi/6. So, I'm going to go with this one:

tan x/2 = sinx/(1+cosx)

that makes the first part of your expression equal to:

tan pi/12 = tan (pi/6)/2 = sin pi/6 / (1+cos pi/6) ...

to find sine and cosine of pi/6 exactly, take out an equilateral triangle (all angles equal to 60) with sides of 2 units. Now, cut it in half. You have a right triangle with angles 60° and 30° (AKA pi/6). The hypotenuse is still equal to 2. The side that got cut in half is equal to 1 now. And the third side is root 3 (pythagorean theorem).

Thus, the sine of pi/6 is 1/2 and the cos pi/6 is (root 3)/2. so that makes 

tan pi/12 = tan (pi/6)/2 = sin pi/6 / (1+cos pi/6) = (1/2)/[1+root(3)/2]. Multiply the numerator and denominator by 2 and you get 1/(2+sqrt3).

NOW! the second part requires a formula for the cosine of a difference of two angles: cos(A−B) = cos A cos B + sin A sin B. So...cos(3pi/2-pi/12) = cos 3pi/2 cos pi/12+sin 3pi/2 sin pi/12. The cos of 3pi/2 is equal to zero. You can check this by drawing a quick graph of y=cosx or by examining the x coordinate of the point on the unit circle which corresponds to 3pi/2 or 270°. SO that gives us sin 3pi/2 sin pi/12. The sin of 3pi/2 is negative one. Again, look at the graph of y=sinx or the y coordinate one the unit circle. Now we have -sin pi/12 for the second half of the expression. Last formula

sin x/2 = sqrt (1-cosB)/2. since pi/12 is half of pi/6 or 30° this is now sqrt (1-cos 30°)/2 or sqrt( 1 - sqrt(3)/2)/2.

So, we have 1/(2+sqrt(3)) - sqrt( 1 - sqrt(3)/2)/2. Multiplying that last fraction by 2 in the numerator and denominator gives the slightly prettier 1/(2+sqrt(3)) - sqrt( 2 - sqrt(3))/4.

Jan 29th, 2015

Studypool's Notebank makes it easy to buy and sell old notes, study guides, reviews, etc.
Click to visit
The Notebank
...
Jan 28th, 2015
...
Jan 28th, 2015
Feb 21st, 2017
check_circle
Mark as Final Answer
check_circle
Unmark as Final Answer
check_circle
Final Answer

Secure Information

Content will be erased after question is completed.

check_circle
Final Answer