Trigonometry

Abstract
This is a review of basic trigonometry and includes several formulas that are
often forgotten if they are not used regularly. Quick derivations of these formulas
are provided in the hopes that by looking over them the reader will be able to
remember how to derive any formula that is forgotten, should the need arise. Thus
a problem in memorization is converted to one of understanding .

1 The Basics
The basic trigonometric functions are sine and cosine. The other four can be obtained
from these two, as follows.

opp is the length of the leg opposite θ
adj is the length of the leg adjacent to θ
hyp is the length of the hypotenuse of the triangle

1.1 Pythagorean Theorem for Trigonometry

Proof Consider a circle of radius r that is centered at the origin. Then a right triangle
is formed by any radius of the circle by drawing a vertical line from the x-axis up to
the point of contact of the radius with the circle. The hypotenuse of this triangle is
known to be r, and the angle is the angle that the hypotenuse makes with the positive
x-axis, as measured counterclockwise. Thus the opposite leg, which is the vertical
displacement, is r sin( θ ), and the adjacent leg, which is the horizontal displacement, is
r cos( θ ). Thus this is now just the normal Pythagorean Theorem for triangles:

Results The above result has two immediate consequences that arise when appropriate
division is performed.

1.2 Negative Angle Formulas

By considering how the sign of the basic sine and cosine functions changes when the
argument is negated (when the radius being considered is reflected across the x-axis),
the fol lowing formulas can be determined.
sin(− θ ) = −sin( θ )
cos(− θ ) = cos( θ )
tan(− θ ) = −tan( θ )
csc(− θ ) = −csc( θ )
sec(− θ ) = sec( θ )
cot(− θ ) = −cot( θ )

Note that the above formulas show, by definition of being an odd or even function,
that sin( θ ), tan( θ ), csc( θ ), and cot( θ ) are odd functions, whereas cos( θ ) and sec( θ )
are even functions.

2 Euler’s Formula and Its Significance

A formula known as Euler’s Formula simplifies the derivation of many trigonometric
identities greatly. It turns the derivations into simple exercises in multiplication of
exponentials. The derivation of this formula is most easily done with calculus, and its
manipulation requires some knowledge of complex numbers ; however, it is worth the
effort. If you are not familiar with calculus, skip the derivation and go directly to how
it can applied in deriving trigonometric identities.

2.1 Euler’s Formula

Proof The proof uses the fact that all three of the functions in the above equation
have simple Taylor series representations, as given below.

Now consider plugging in x = iθ in the Taylor series for e^x. Remembering that
i² = −1, this yields the following equation, where the real and imaginary parts have
been separated.

Now it will be noticed that the two infinite series in parentheses are just the Taylor
series for sin( θ ) and cos( θ ). Since these series are universally convergent, this means
that the following equation holds, as was to be shown.

In order to make much use of Euler’s Formula, an additional formula must be articulated,
for the sake of clarity.

2.2 DeMoivre’s Theorem
(cos( θ ) + i sin( θ ))ⁿ = cos(nθ ) + i sin(nθ )

Explanation DeMoivre’s Theorem may seem to be coming out of nowhere, but consider
Euler’s Formula and the laws of exponents. When viewed in that light, all that
DeMoivre’s Theorem is saying is that which is clearly valid if the
ordinary laws of exponents hold for complex numbers. The ordinary laws of exponentiation
do, in fact, still hold. So just trust your instinct when manipulating Euler’s
Formula, and you will not run into trouble if you are using the laws of exponentiation
appropriately.

2.3 Angle Addition,
Angle Subtraction, and Double Angle Formulas
The subtraction and double angle formulas are just specific cases of the addition formulas.
The subtraction formulas are obtained by replacing y with −y and using the
negative angle formulas. The double angle formulas are obtained by setting x = y.
Thus all of these results are summarized by the following equations.
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

Proof of Angle Addition Formulas Both angle addition formulas can be proven at
once by using Euler’s Formula in combination with DeMoivre ’s Theorem.

e^ixe^iy = e^i(x+y)
[cos(x) + i sin(x)][cos(y) + i sin(y)] = cos(x + y) + i sin(x + y)
[cos(x) cos(y)−sin(x) sin(y)]+i[sin(x) cos(y)+cos(x) sin(y)] = cos(x+y)+i sin(x+y)

Thus, equating real and imaginary components, the desired result is obtained.

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
cos(x + y) = cos(x) cos(y) − sin(x) sin(y)

Use Specific Values The angle addition formulas are incredibly versatile in terms
of deriving other useful identities on the fly. For example, it is easily seen that

Remember the Derivation The derivation of the angle addition formulas is worth
remembering. If you are in need of a triple angle formula, for example, it is simple
enough to calculate and equate the real and imaginary components, as
was done in the derivation above. Thus, without the need for any memorization, one
technique enables you to quickly derive a great variety of trigonometry formulas. It is
very useful.

3 Sine and Cosine Laws

Two other widely known trigonometry formulas that do not follow from the above
techniques are the Law of Sines and the Law of Cosines.

3.1 Law of Sines
Consider a triangle with side lengths a, b, and c, with opposite angles A, B, and C,
respectively. Then the following equations hold.

Proof If it can be shown that b sin(A) = a sin(B) and b sin(C) = c sin(B) for
any triangle, then the triple equality will follow immediately. When ex pressed in this
fashion , however, it is clear that these 2 equalities always hold. The first expression
is an equality between two ways of expressing the altitude of the triangle from side
c up to the vertex containing angle C. The second expression is an equality between
two ways of expressing the altitude of the triangle from side a to the vertex containing
angle A. Thus the proof is complete.

3.2 Law of Cosines
Again consider a triangle with side lengths a, b, and c, with opposite angles A, B, and
C, respectively. Then the following equations hold (one basic equation that is valid for
each of 3 possible set-ups for any triangle).
a² = b² + c² − 2bc cos(A)
b² = a² + c² − 2ac cos(B)
c² = a² + b² − 2ab cos(C)

Proof There is one fundamental equation that must hold regardless of which vertex
contains the angle that is being singled out. Thus, without loss of generality, consider
the vertex containing angle C.

Two edges are going out from this vertex, namely those of length a and b. Consider
these edges to be vectors and of length a and b, respectively, going out from the
vertex containing angle C.

The opposite edge, of length c can then be expressed as a vector in two ways:
This is of no concern, however, because all we’re really dealing
with is a triangle. Thus all we care about is which is the same for both definitions
because the vectors are equal in magnitude, just opposite in direction. Similarly,
is unaffected by our choice of .

Now consider Using the definition for specificity, this can be
calculated using the properties of the dot product and distributivity . Note also that the
angle between the vectors and is C.

Thus the desired result has been obtained, and the proof is complete.