Trigonometry Review
Fourteen Trigonometric Identities
We present here 14 trigonometric identities that every mathematics
student should know with perfect accuracy. Memorize some of them, and
know how to quickly derive the others from the ones you memorize.
The first four identities are basic.
2 2
1. sin x + cos x = 1
2. sin(-x) = -sin(x)
3. cos(-x) = cos(x)
4. sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
The next three identities follow easily from identities #2, #3, and #4.
To derive #6, differentiate both sides of #4 with respect to one of
the variables.
5. sin(a - b) = sin(a)cos(b) - sin(b)cos(a)
6. cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
7. cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
From #4 and #6 the next two identities are easily derived.
8. sin(2x) = 2sin(x)cos(x)
2 2
9. cos(2x) = cos (x) - sin (x)
From #9, the next two identities follow.
1 + cos(2x) 2
10. ------------- = cos (x)
2
1 - cos(2x) 2
11. ------------- = sin (x)
2
Next come the half angle formulas, which follow from #10 and #11.
2 1 + cos(a)
12. cos (a/2) = ------------
2
2 1 - cos(a)
13. sin (a/2) = ------------
2
2
Finally, #14 follows by dividing both sides of #1 by cos (x).
2 2
14. tan (x) + 1 = sec (x)
Exact values of trigonometric functions
Every mathematics student should know the following table by heart.
x sin(x) cos(x) tan(x) sec(x)
0 0 1 0 1
pi/6 1/2 (sqrt(3))/2 1/(sqrt(3)) 2/(sqrt(3))
pi/4 1/(sqrt(2) 1/(sqrt(2) 1 sqrt(2)
pi/3 (sqrt(3))/2 1/2 sqrt(3) 2
pi/2 1 0 undefined undefined
Definitions of the Trigonometric Functions
Each mathematics student should be able to define the trigonometric
functions. A concise definition follows.
An angle in the coordinate plane is in standard position if its vertex
coincides with the origin, (0,0), and its initial side coincides with
the positive x-axis. For such an angle, T, let P(a,b) be a point
other than (0,0) on its terminal side. Let r be the distance from P
to the origin. Note that by the Pythagorean Theorem,
2 2
r = sqrt( a + b ) .
The functions sin(T) and cos(T) are now defined as follows:
b a
sin(T) = --- and cos(T) = ---
r r
The other four trigonometric functions are defined in terms of 'sin'
and 'cos' as follows:
sin(T) cos(T)
tan(T) = ------ cot(T) = ------
cos(T) sin(T)
1 1
sec(T) = ----- csc(T) = -----
cos(T) sin(T)
An angle is often represented by its radian measure or its measure in
degrees. It is important to use radians whenever derivatives are
involved, because otherwise the familiar formulas for derivatives of
trigonometric functions are not valid.
When dealing with a triangle, it is customary to consider all the angles to
be between 0 radians and pi radians. Two fundamental tools for solving
general triangles are the law of sines and the law of cosines.
LAW OF SINES and LAW OF COSINES
Let the lengths of the three sides of a triangle be denoted a, b,
and c. Let the angle opposite side a be u, and the angle opposite
side b be v and the angle opposite side c be w.
sin u sin v sin w
LAW OF SINES: ------- = ------- = -------
a b c
2 2 2
LAW OF COSINES: c = a + b - 2ab cos w
Exercises
1. Show that sin(pi/5) = sin(4pi/5).
2. Express sin(4pi/5) in terms of sin(pi/5) and cos(pi/5).
3. Show that cos(pi/5) is a root of the polynomial 8x^3 - 4x - 1.
4. Show that the only positive root of the polynomial in problem 3
is
sqrt(5) + 1
-------------
4 ,
and conclude that this expression is cos(pi/5).
5. Give an expression for the exact value of sin(pi/5).
6. Expand sin(pi/5 + 4pi/5) to get an equation of the form:
0 = [expression in sin(pi/5) and cos(pi/5)]
7. Simplify the equation from problem 7, by cancelling out a common factor
on the right.
8. Get the equation from 8 into the form 0 = [ a quadratic expression in
either (sin(pi/5))^2 or (cos(pi/5))^2].
9. Solve that quadratic equation, and do the obvious things to find
expressions for sin(pi/5) and cos(pi/5). Problems 6 - 9 constitute
an alternate derivation of the information in problems 5 and 6.