SOLUTION: Use identities to write the expression as a function of x. tan(x+pi/6)

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Question 1204959: Use identities to write the expression as a function of x.
tan(x+pi/6)
Found 3 solutions by MathLover1, ikleyn, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
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tan%28x%2Bpi%2F6%29
use identities:

tan%28x%2Bpi%2F6%29=sin%28x%2Bpi%2F6%29%2Fcos%28x%2Bpi%2F6%29
sin%28a%2Bb%29=cos%28b%29+%2Asin%28a%29+%2B+cos%28a%29%2A+sin%28b%29...a=x, b=pi%2F6
sin%28x%2Bpi%2F6%29=cos%28pi%2F6%29+%2Asin%28x%29+%2B+cos%28x%29%2A+sin%28pi%2F6%29
cos%28a%2Bb%29=cos%28a%29%2A+cos%28b%29+-+sin%28a%29+%2Asin%28b%29
cos%28x%2Bpi%2F6%29=cos%28x%29%2A+cos%28pi%2F6%29+-+sin%28x%29+%2Asin%28pi%2F6%29
then we have



since cos%28pi%2F6%29=sqrt%283%29%2F2+ and sin%28pi%2F6%29=1%2F2









Answer by ikleyn(52906) About Me  (Show Source):
You can put this solution on YOUR website!
.
Use identities to write the expression as a function of x.
tan(x+pi/6)
~~~~~~~~~~~~~~~~~~~ 

Use trigonometric identity - the addition formula for tangents

    tan(a+b) = %28tan%28a%29%2Btan%28b%29%29%2F%281-tan%28a%29%2Atan%28b%29%29.    (1)


It gives

    tan%28x+%2B+pi%2F6%29 = %28tan%28x%29+%2B+tan%28pi%2F6%29%29%2F%281+-+tan%28x%29%2Atan%28pi%2F6%29%29.    (2)


Next, use the formula of half-angle for tangents to express  tan%28pi%2F6%29

    tan%28pi%2F6%29 = tan%28%28%28pi%2F3%29%29%2F2%29 = sin%28pi%2F3%29%2F%281%2Bcos%28pi%2F3%29%29%29 = %28%28sqrt%283%29%2F2%29%29%2F%281%2B1%2F2%29 = %28%28sqrt%283%29%2F2%29%29%2F%28%283%2F2%29%29 = sqrt%283%29%2F3.    (3)


Continue (2)

    tan%28x+%2B+pi%2F6%29 = %28tan%28x%29+%2B+tan%28pi%2F6%29%29%2F%281+-+tan%28x%29%2Atan%28pi%2F6%29%29 = %28tan%28x%29+%2B+sqrt%283%29%2F3%29%2F%281-tan%28x%29%2A%28sqrt%283%29%2F3%29%29 = %283%2Atan%28x%29%2Bsqrt%283%29%29%2F%283-sqrt%283%29%2Atan%28x%29%29.


ANSWER.  tan%28x%2Bpi%2F6%29  simplified is  %283%2Atan%28x%29%2Bsqrt%283%29%29%2F%283-sqrt%283%29%2Atan%28x%29%29.

Solved



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutors mathlover and ikleyn have great answers.
Despite their final answers looking completely different, they are actually equivalent.

To go from mathlover's answer to ikleyn's answer, multiply top and bottom by sqrt%283%29%2Fcos%28x%29

sqrt%283%29%2Asin%28x%29+%2B+cos%28x%29
in the numerator becomes
where tan%28x%29+=+sin%28x%29%2Fcos%28x%29



sqrt%283%29%2Acos%28x%29-+sin%28x%29
in the denominator becomes



Therefore we can claim that
is an identity.
A graphing tool can be used to confirm that the two sides are the same.

Desmos Graph
https://www.desmos.com/calculator/av7ncyhcj5
Note how one curve perfectly overlaps the other.

It is a true equation for all x in the domain.
That domain being x is the set of all real numbers but x+%3C%3E+pi%2F3+%2B+pi%2An where n is an integer.
If x+=+pi%2F3%2Bpi%2An was the case, then it causes a division by zero error.