Essential Trigonometric Functions & Derivatives
These trig derivatives and properties have been prevalent in the homework. It’s likely that
you’ll see them on the exam as well.
Properties mentioned in the homework
1. The basics:
tan(x) =
sin(x)
cos(x)
csc(x) =
1
sin(x)
1
cos(x)
=
tan(x)
sin(x)
1
sec(x) =
cos(x)
cot(x) =
2. Sine and cosine are bounded. This means that for all real numbers x,
−1 ≤ sin(x) ≤ 1
and
− 1 ≤ cos(x) ≤ 1
3. For any real number a, the function
f (x) = sin(a/x)
is not continuous at zero, because the curve oscillates between -1 and 1 with increasing
frequency as x approaches zero. Recall from the homework though, that the function
f (x) = x2 sin(a/x)
is both continuous and differentiable when we specify f (0) = 0. Make sure you understand why.
4. Near zero, the graphs of f (x) = x and f (x) = sin(x) are almost identical. This means
that sin(x) looks like a straight line through zero, and that:
sin(x)
=1
x→0
x
lim
The same is true for f (x) = sin(ax) for any real number a. Near zero, sin(ax) ≈ ax
and
sin(ax)
lim
=1
x→0
ax
Another common limit (which can be derived using the previous one) is:
cos(x) − 1
=0
x→0
x
lim
1
5. Here are some common values of sin, cos, and tan you should be familiar with:
degrees
0
30
45
60
90
180
270
radians
0
π/6
π/4
π/3
π/2
π
3π/2
sin(x)
0
1/2
√
2/2
√
3/2
1
0
-1
cos(x)
1
√
3/2
√
2/2
1/2
0
-1
0
tan(x)
0
√
3/3
1
√
3
undefined
0
undefined
Derivatives
I recommend memorizing these derivatives.
1.
d
dx
sin(x) = cos(x)
2.
d
dx
cos(x) = − sin(x)
3.
d
dx
tan(x) = sec2 (x)
4.
d
dx
cot(x) = − csc2 (x)
5.
d
dx
sec(x) = sec(x) tan(x)
6.
d
dx
csc(x) = − csc(x) cot(x)
Common Identities
Occasionally you need to know some trig identities to simplify a problem. Here are a few of
the most common identities.
1. Even-Odd Identities.
sin(−x) = − sin(x)
cos(−x) = cos(x)
tan(−x) = − tan(x)
2. Pythagorean Identities.
sin2 (x) + cos2 (x) = 1
tan2 (x) + 1 = sec2 (x)
cot2 (x) + 1 = csc2 (x)
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3. Sum and Difference Formulas.
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
sin(x − y) = sin(x) cos(y) − cos(x) sin(y)
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
cos(x − y) = cos(x) cos(y) + sin(x) sin(y)
Please refer to the appendix of your textbook, page A28, for more trig identities, such
as the half-angle and double-angle formulas.
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