Trigonometry Page

Rradians and Degrees:
For reasons not totally relavant for simple trigonometry, there exists a unit of angular measure called radians. This has to do with arc length, and as a bonus, it makes calculus a bit easier. For a unit circle (a circle with radius 1), one radian sweeps out an arc of length 1.

Conversion:
1 degree = Pi/180 radians
1 radian = 180 degrees / Pi
Pi radians = 180 degrees; Pi/2 radians = 90 degree
s = r*t, where r is the raduis of a circle, t is the angle of a swept arc, and s is the length of the arc swept out.

The fundemental funtion to trigonometry is sin(t). This gives a sine wave. From it are based 5 other functions: an offset, a division and 3 inverses. For this sheet, t is a variable defining angle.
This page's Bracket standard: {[()]} The parantheses are the innermost, square brackets are for second innermost, and so on for {}.

Identities:
sin^2(t) + cos^2(t)= 1
cos(-t) = cos(t)
csc(t) = 1/sin(t)
sec(t) = 1/cos(t)
tan(t) = sin(t)/cos(t)
cot(t) = 1/tan(t) = cos(t)/sin(t)

Trig Functions of Useful Angles

t Degrees  radians    sin(t)       cos(t)       tan(t)
--------------------------------------------------------
    0        0          0            1            0
   30       Pi/6       1/2        SQRT(3)/2    SQRT(3)/3     
   45       Pi/4     SQRT(2)/2    SQRT(2)/2       1
   60       Pi/3     SQRT(3)/2      1/2         SQRT(3)
   90       Pi/2        1            0          invalid

The 'invalid' entry is because the tan(90) is undefinded, where sin(90)/cos(90) = 1/0, which is division by zero, an undefined or infinite state. Which direction infinite depends on the application.

Law of Sines:
sin(A)/a + sin(B)/b + sin(C)/c
where you have a triangle with sides a,b,c, and the angle opposite a is A, and so on for B & C.

Law of Cosines:
a^2 = b^2 + c^2 - 2bc*cos(A)
b^2 = a^2 + c^2 - 2ac*cos(B)
c^2 = a^2 + b^2 - 2ab*cos(C)

Double - Angle Formulas
sin(2*t) = 2*sin(t)*cos(t)
cos(2*t) = cos^2(t) - sin^2(t) = 2*cos^2(t) -1 = 1 - 2*sin^2(t)
tan(2*t) = 2*tan(t) / [1-tan^2(t)]

Half - Angle Formulas
sin^2(t) = [1 - cos(2*t)] / 2
cos^2(t) = [1 + cos(2*t)] / 2

Addition and Subtraction 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)
tan(x+y) = [tan(x)+tan(y)] / [1-tan(x)*tan(y)]
tan(x-y) = [tan(x)-tan(y)] / [1+tan(x)*tan(y)]

exponential equivalents:
sin(t) = (e^it - e^-it)/2i
cos(t) = (e^it + e^-it)/2
tan(t) = (e^it - e^-it) / i(e^it + e^-it)
where i = SQRT(-1), the unit imaginary number.

Disclaimer:

Contact me at jdemaa17@calvin.edu

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This page was last updated created September 17, 1996.