Wikipedia:Reference desk/Archives/Mathematics/2015 May 14

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May 14

easy formulas to calculate sin(), cos() and other functions.

Hello. In a mathematic book that I have, there are about 30 formulas to calculate trigonometric functions. Some of them are:

sin(2*pi - x) = sin(x)
cos(2*pi - x) = cos(x)
sin(2*pi  + x) = sin(x)
cos(2*pi + x) = cos(x)
sin(pi - x)= sin(x)
cos(pi - x)= -cos(x)
sin(pi + x) = -sin(x)
cos(pi + x) = -cos(x)
sin(pi/2 - x)= cos(x)
cos(pi/2 - x) = sin(x)
sin(pi/2 + x) = cos(x)
cos(pi/2 + x)= -sin(x)
sin(-x) = -sin(x)
cos(-x) = cos(x)

and many other similar formulas for tan() and cot(). The question I have is that these formulas are hard to remember and one may forget these formulas in a long time, I'm looking for an easy way to remember these formulas. Is there an easy way to calculate these values, say, sin(5 * (pi/6)) or cos(5 * (pi/6))? Can I just calculate sin() or cos() of (pi/6), and then specify the sign of the result according to the coordinates of the point? For example, 5 * (pi/6) is in the second quarter of the unit circle and in this quarter, sin() is positive and cos() is negative, so the result of sin(5 * (pi/6)) would be (1/2) and cos(5 * (pi/6)) would be (-sqrt(3)/2). Am I correct? 46.224.147.50 (talk) 17:56, 14 May 2015 (UTC)

Yep, you got it. See e.g. Special_right_triangles#Angle-based - once you memorize the values of sin/cos for the special angles <pi/2, you can get any multiple you like, as long as you get the sign right by thinking of the unit circle and quadrants, as you describe. Note also the pattern: ${\frac {\sqrt {0}}{2}},{\frac {\sqrt {1}}{2}},\ldots {\frac {\sqrt {4}}{2}}$ is the same for sin and cos, just reversed. Then of course you can get sec, tan, etc from their identities in terms of sin/cos. Honestly, that's all the memorization of trig stuff I've ever had to keep. Sure, double angle formulae are nice on occasion, as well as other more exotic stuff, but for those, it's good enough just to remember that they exist and can be looked up. (unless you are a student currently being taught these identities in an algebra/trig class, then you may well have to pull them from memory for a test, though such demands are fortunately less common than than they used to be :) SemanticMantis (talk) 18:09, 14 May 2015 (UTC)

Once you know the sines and cosines of the usual angles of 0, 30, 45, 60 and 90 degrees, another way of getting that of any other such as 210 or 315 degrees is from a quick sketch (possibly just visualised) of the functions between 0 and 360 - which incidentally would show you that your first formula [sin(2*pi - x) = sin(x)] is wrong.→109.145.13.182 (talk) 19:30, 14 May 2015 (UTC)

Note that you are missing a negative sign in your first formula, which should read: sin(2*pi - x) = -sin(x). ToE 21:33, 14 May 2015 (UTC)

It's enough to remember the following formulas:

sin(pi + x) = -sin(x)
cos(pi + x) = -cos(x)
sin(pi/2 - x)= cos(x)
sin(-x) = -sin(x)
cos(-x) = cos(x)

All the others follow from them trivially. Also, these formulas will be easier to remember if you understand them. With enough experience working with trigonometric functions, looking at their graphs, etc., these formulas will become obvious. -- Meni Rosenfeld (talk) 21:54, 14 May 2015 (UTC)

I heartily agree with MR, and suggest that you learn to "translate" his equations plus a few others from your list into words which describe some properties of the sine and cosine functions which you should already be familiar with:

• sin(2*pi + x) = sin(x) & cos(2*pi + x) = cos(x) : Sine and cosine are periodic functions with period 2*pi.
• sin(pi + x) = -sin(x) & cos(pi + x) = -cos(x) : Translating either function by half a period is equivalent to reflecting it across the horizontal axis, also know as negating it. (Hmm, it seems as if there should be a name for this property.)
• sin(pi/2 - x)= cos(x) & cos(pi/2 - x)= sin(x) : The sine of an angle is equal to the cosine of its complement and vice versa.
• sin(pi/2 + x) = cos(x) : The cosine function is equivalent to the sine function phase shifted ahead (translated left) by one quarter period.
• sin(-x) = -sin(x) & cos(-x) = cos(x) : Sine is and odd function and cosine is an even function.

ToE 04:31, 15 May 2015 (UTC)