Download Trigonometric Identities Special Angles

Trigonometric Identities
There are a few trigonometric identities which we must learn to identify on
sight.
First, we have the Pythagorean identities:
• sin2 (x) + cos2 (x) = 1
• tan2 (x) + 1 = sec2 (x)
• 1 + cot2 (x) = csc2 (x)
Then the double angle formula for sine and for cosine.
• sin(2x) = 2 sin(x) cos(x)
• cos(2x) = cos2 (x) − sin2 (x)
Combining the double angle formula for cosine with the first Pythagorean
identity, we get a “square elimination” formula for both sine and cosine. This
is vital for integrating the left side of each identity.
• sin2 (x) = 21 (1 − cos(2x))
• cos2 (x) = 21 (1 + cos(2x))
There are clean expressions for things like
sin(x + y)
but we use them rarely enough that they are generally not worth memorizing.
Even the third Pythagorean identity is used quite rarely, but does come up a
few times.
The main point with memorizing these particular identities is a recognition
question. For several types of calculus problems, it must occur to you at a certain
point that one of these identities would simplify your work, or you simply cannot
continue with the problem. Without them, you would simply stare at a blank
page or waste your time doing something which cannot succeed.
Special Angles
The values of the trigonometric functions at the so-called special angles come
up often enough in Calculus that we need to memorize them. You can simply
memorize the table below or you could remember the trick described on the
next page.
x
sin(x)
cos(x)
tan(x)
0
0
1
0
π
√4
2
√2
2
2
π
6
1
2
√
3
2
1
√
3
1
1
π
√3
3
2
1
√2
3
π
2
1
0
DNE
The trick is that we can do a little bit of rewriting to make remembering the
table much easier. You can see that we have
√
√
√
0 1
1
4
0=
, =
and 1 =
2 2
2
2
If we rewrite things this way, the first two lines of the chart become:
x
sin(x)
cos(x)
0
√
0
√2
4
2
π
√6
1
√2
3
2
π
√4
2
√2
2
2
π
√3
3
√2
1
2
π
√2
4
√2
0
2
√
In this form, we just have to remember that each entry looks like 2? where the
? is a counting number from zero to four. You can even keep track of that
number from zero to four using your fingers, and I’ll show you that in class.
To work out the tangent of an angle this way, say π6 , we remember that
tan
π
6
sin( π6 )
=
=
cos( π6 )
√
1
√2
3
2
r
=
1
3
In other words, every special value of tangent will be the square root of a
fraction. The pieces of the fraction can be read off from the numerators of the
sine and cosine of that angle.
2