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Trigonometry
If you took a course in trigonometry, and spent hours on the many esoteric minutiae
that it can lead to, I want to assure you that the number of those concepts that you use
in day-to-day work is small. But, you use those every day: The definitions of the sine,
cosine, and tangent. A few of the identities among them. Mostly, how to recognize them in
unfamiliar contexts.
The basis of trigonometry lies in the fact that for similar triangles, the ratio of corresponding sides is the same. Remember that similar triangles have the same shape (not
necessarily the same size), so that their angles are equal. The trigonometric names are simply
the titles given to these ratios in the particular case of a right triangle.
c'
c
θ
a
a'
b
b'
The definitions are
sin(θ) =
a
a0
= 0,
c
c
cos(θ) =
b
b0
= 0,
c
c
tan(θ) =
a
a0
= 0.
b
b
The Greek letter θ (theta) is commonly used for angles, as is the Greek letter φ (phi). The
other definitions, such as the cotangent: cot θ = 1/ tan θ show up too, but less often.
From the definitions, you immediately get an identity:
tan θ =
a
a/c
sin θ
=
=
.
b
b/c
cos θ
Similarly, start from the Pythagorean theorem, a2 + b2 = c2 , and divide both sides by c2 :
a2 + b2 = c2 =⇒
a2 b 2
+
= 1,
c2 c2
or
sin2 θ + cos2 θ = 1.
There are a few other common identities that take a little more work to derive, such as
sin(θ + φ) = sin θ cos φ + cos θ sin φ,
and
cos(θ + φ) = cos θ cos φ − sin θ sin φ.
The values of these functions at simple angles such as 0, 30◦ , 45◦ , 60◦ , and 90◦ are ones
that you should know (and in fact they are so easy to derive from the definitions that you
should be able to derive them). Just draw a couple of pictures to see what they are. An
equilateral triangle, divided in two and an isosceles right triangle are all that it takes.
1
2
Trigonometry
30
ο
60
o
45
o
Half the vertex angle of the equilateral triangle is 30◦ , and the side opposite the 30◦
angle is half the full side, so immediately the sine of 30◦ is 1/2. Also the same picture shows
that the cosine of 60◦ is the same thing, 1/2. The two angles of the isosceles right triangle
are 45◦ , and the two sides are equal, so the tangent of 45◦ is obviously 1. To get the sine
of 45◦ for example,
you use the Pythagorean theorem to find the hypotenuse, then you find
√
sin 45◦ = 1/ 2.
Trigonometry
Practice Set 1
[1] From the second picture of the introduction, derive
the tangent of 30◦
[2] In the formula for the sine of the sum of two angles,
set θ = φ and derive the expression for sin(2θ).
[3] A tree is shaped as an isosceles triangle with a vertex
angle of 30◦ . Its base is 1.00 meters. What is its
height?
[4] The shadow of a tower standing on a horizontal plane
is found to be 60 m longer when the sun’s altitude is
30◦ than when it is 45◦ . Find the height of the tower.
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4
Trigonometry
Solutions for Practice Set 1
[1] From the second picture of the introduction,
derive the tangent of 30◦
If the side is L, the height comes
p from the
√
Pythagorean theorem: It is L2 − (L/2)2 = L 3/2.
The tangent of 30◦ is then the tangent of half the
L/2
1
vertex angle, which is √
=√ .
L 3/2
3
[2] In the formula for the sine of the sum of two
angles, set θ = φ and derive the expression for
sin(2θ).
sin(θ + θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ.
[3] A tree is shaped as an isosceles triangle with a
vertex angle of 30◦ . Its base is 1.00 meters. What
is its height?
Half the vertex angle is 15◦ . The
tangent of this angle is half the base over
the unknown height. tan 15◦ = 0.500 m/h,
so h = 0.500 m/ tan 15◦ = 1.87 m.
15
ο
1m
[4] The shadow of a tower standing on a horizontal
plane is found to be 60 m longer when the sun’s
altitude is 30◦ than when it is 45◦ . Find the
height of the tower.
h
h
−
◦
tan 30
tan 45◦
= 60 m.
h
Solve
for h to get
.
h = 60 m (cot 30◦ − cot 45◦ ) = 82 m.
60 m
Trigonometry
Practice Set 2
[1] The length of a kite string is 250 yards, and the angle
of elevation of the kite is 40◦ . Find the height of
the kite, supposing the line of the kite string to be
straight.
[2] From the second picture of the introduction, derive
the tangent of 60◦
[3] A chimney stands on a horizontal plane. At one point
in this plane the angle of elevation of the top of the
chimney is 30◦ , at another point 100 feet nearer the
base of the chimney the angle of elevation of the top
is 45◦ . Find the height of the chimney.
[4] At a point midway between two towers on a horizontal plane the angles of elevations of their tops are
30◦ and 60◦ respectively. What is the ratio of their
heights?
[5] What are the cosine and the sine of π/2?
5
6
Trigonometry
Practice Set 3
[1] One of the equal sides of an isosceles triangle is 50 cm
and one of its equal angles is 40◦ . Find the base, the
altitude, and the area of the triangle.
[2] What is cos θ tan θ?
[3] From the Pythagorean theorem, a2 + b2 = c2 , Divide
by a2 and what identity do you get? Same for b2 .
[4] Check the identity for the sine of the sum of two
angles by trying the specific angles θ = 30◦ and φ =
60◦ .
[5] Solve the equation sin 2θ = cos θ.