18.01 Section, September 21, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Trig, polar coordinates, and chain rule
r
1. Differentiate
1 − sin x
.
cos3 x
2. Consider a hexagon inscribed in a circle, as in the picture. Forget that you probably know
that the triangles are equilateral; part of this problem is to show that.
(a) What is the sum of all the interior angles (marked θ)?
(b) What is the side length of the hexagon?
3. In the diagram below, the circle is a unit circle, and the line that looks like a tangent line
is actually a tangent line. Find the equation of the tangent line and find x.
1
Hint: if a line has slope m, lines perpendicular to it have slope − m
.
1
4. Differentiate the equality tan(tan−1 (x)) = x to get an expression for the derivative of
tan−1 (x). Simplify your answer so it contains no trig functions.
5. Practice converting between polar and rectangular coordinates:
(a) Convert to rectangular coordinates: r2 = r cos θ. What kind of shape is it?
(b) What do the graphs of the polar functions θ = c and r = c look like?
(c) What does the graph of r = cos θ look like?
6. Bonus question: Come up with an implicit function (i.e. something of the form f (x, y) = 0
instead of y = f (x)) describing what you get when you take the parabola y = x2 and rotate
it 45◦ clockwise.
Hint: first convert to polar coordinates, do the rotation, then convert back to rectangular.
2
Review
• Inverse trig functions:
1
sec x =
cos x
• Sum formulas:
csc x =
1
sin x
cot x =
1
tan x
tan x =
sin x
cos x
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
• Double angle formulæ:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ
= 1 − 2 sin2 θ = 2 cos2 θ − 1
2 tan θ
tan 2θ =
1 − tan2 θ
• Half-angle formulæ:
r
1 − cos θ
2
1 + cos θ
cos( 12 θ) = ±
2
r
1 − cos θ
tan( 12 θ) = ±
1 + cos θ
1 − cos θ
sin θ
=
=
sin θ
1 + cos θ
sin( 12 θ) = ±
r
• Chain rule:
d
f (g(x)) = f 0 (g(x)) · g 0 (x)
dx
• Trigonometric derivatives
d
d
sin x = cos x
cos x = − sin x
dx
dx
d
d
tan x = sec2 x
cot x = − csc2 x
dx
dx
d
1
d
−1
sin−1 x = √
cos−1 x = √
2
dx
dx
1−x
1 − x2
• Converting rectangular coordinates to polar coordinates:
x = r cos θ
d
1
tan−1 x = 2
dx
1x
y = r sin θ
• Converting polar coordinates to rectangular coordinates:
p
r = x2 + y 2
θ = tan−1
3
y
x