# Tan 6x is exactly the same thing as 6 tan x

1. Nov 8, 2013

### skj91

Just want to verify if I am correct in assuming that

tan 6x is exactly the same thing as 6 tan x

It's merely a matter of rearranging the variables of algebra's multiplicative identity, am I right?

2. Nov 9, 2013

### Staff: Mentor

No thats not right.

You can test this easily with your calculator:

Does 6 * tan (10 degrees) =?= tan (60 degrees) ?

You have to use the tan(a+b) formula to break it down the right way:

tan(a+b) = ( tan(a) + tan(b) ) / ( 1-tan(a)*tan(b) )

3. Nov 9, 2013

### Staff: Mentor

I would guess you're thinking of multiplication associativity, which says that a(bc) = (ab)c. The trouble is, and this is a common mistake, tan 6x does NOT mean tan * 6 * x. I usually use parenthese to emphasize the fact that tan represents a function, not a factor in a product.

4. Nov 9, 2013

### Student100

Dose that actually work? I've never seen it as a sum before.

I assumed tan(nx) was only represented by continuous fractions:

$\frac {ntan(x)}{1- \frac {(n^2-1^2)tan^2(x)}{...}}$

5. Nov 9, 2013

6. Nov 9, 2013

### Student100

I feel dumb looking at all those trig identities, I've seen the product to sum and sum to product for sine and cosine, but not for tangent. Honestly I'm still lost by your example, can you show how you break tan(6x) into a sum? I think I could follow from there.

7. Nov 9, 2013

### Staff: Mentor

Okay tan(6x) = tan(3x + 3x) = ( tan(3x) + tan(3x) ) / ( 1 - tan(3x)*tan(3x) )

So now break down tan(3x) = tan(2x + x) and repeat the process...

8. Nov 9, 2013

### Student100

Wow.... I think I need another cup of coffee. Missing the forest for the trees. Thanks Jedi.

9. Nov 13, 2013

### skj91

So basically I need to view the internals of any pie functions not as some value with substance, like a measurement or quantity, but instead see it as an indicator of what actions to perform on those contents, like a pattern of operations or protocol? And the values given will provide sort-of a starting point or clue as to which operations--which sine or cosine laws or rules--are applicable? Because straight across algebra isn't correct? Unless the values given are in radians or degrees, right? Then you can treat them like normal algebra, right?

10. Nov 13, 2013

### Staff: Mentor

It seems you are over-thinking this or trying to generate some new rule that may be too complicated to guide your mathematical life.

For me I just remember that when working with trig functions, I have a bunch of identities I can use to simplify them and that I need to use them in conjunction with algebra to solve the problem. Its similar for log and power functions where I try to remember the associated log/power identities.

When I saw the tan(6x) it triggered something in my brain to say I need to reduce this to something in x in order to solve for x and the only thing that came to mind was the tan(a+b) identity.

Last edited: Nov 13, 2013
11. Nov 13, 2013

### Staff: Mentor

"pie functions" ?? If you're referring to $\pi$, that's "pi." Pie is something to eat; pi is a number.
These are trig functions.
There are six trig functions: sin, cos, tan, cot, sec, and csc. The tan, cot, sec, and csc functions can be rewritten in terms of sin or cos or both. These are identities that you learn when you're starting with trig.

There are also addition formulas for all six, but the most important are sin(A + B) and cos(A + B).

From the addition formulas, it's easy to derive the multiple angle identies, such as sin(2A) and cos(2A) and others.
What you were doing wasn't "straight across algebra" since you were ignoring the fact that tan(6x) does NOT mean tan * 6 * x. It means the tangent of 6x, in the same way that √(6x) means the square root of 6x.
Doesn't make any difference. If you had tan(6 * 30°) that would be the tangent of 6 * 30°, or the tangent of 180°.
???