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Notes on Plane Trigonometry

Plane trigonometry deals with the relations between the angles and sides of triangles of which the three vertices are located on the surface of a plane and the sides are straight lines (a straight line being the shortest connection between two points on the plane).
A fundamental characteristic of plane euclidian geometry is that the sum of angles of an arbitrary triangle on the surface of the plane equals 180°. This is in contrast to triangles on elliptic or hyperbolic surfaces for which the sum of angles are larger or smaler than 180°.

Types of Angles

  • An angle equal to 90° or PI/2 radians is called a right angle.
  • An angle equal to 180° or two right angles is called straight angle.
  • Angles that are not right angles or a multiple of a right angle are called oblique angles.
  • Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
  • Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
  • Two angles that sum to one right angle (90°) are called complementary angles.
  • Two angles that sum to a straight angle (180°) are called supplementary angles.
  • Two angles that sum to a full circle (360°) are called explementary angles or conjugate angles.

The Plane Oblique Triangle

An arbitrary plane triangle is defined by three sides with lengths S1, S2 and S3 and the three related vertex angles a1, a2 and a3. The sum of the interior vertex angles a1, a2 and a3 is 180°.

Law of Cosines for Sides:

  S12 = S22 + S32 - 2 * S2 * S3 * cos(a1)
  S22 = S12 + S32 - 2 * S1 * S3 * cos(a2)
  S32 = S12 + S22 - 2 * S1 * S2 * cos(a3)

Law of Sines:

  sin(a1)/S1 = sin(a2)/S2 = sin(a3)/S3

Law of Cosines for Angles:

  cos(a1) = (S32 + S22 - S12) / (2 * S2 * S3)
  cos(a2) = (S32 + S12 - S22) / (2 * S1 * S3)
  cos(a3) = (S22 + S12 - S32) / (2 * S1 * S2)

Basic Trigonometric Functions


The basic trigonometric functions sin(x) and cos(x) are directly related to the basic relationships between angles and sides of plane right triangles.

In an othogonal cartesian grid defined by two orthogonal axis X and Y, angles are measured refering to the horizonal X axis. Each point "A" on the unity circle, defines an angle "a" as well as a right triangle with the sides OA, AP and PO with the intersecting angles a, 90°-a and 90°.

Each of these triangles defines the sin(a) and cos(a) as the orthogonal sides of the triangle. The third side is called the hypotenuse. By construction, the length of this hyponenuse is 1 (the radius of the unity circle). Applying Pythagoras's law for right triangles, yields the following basic identity:

sin2(x) + cos2(x) = 1

Fundamental Identities

The following general identities can be easily verified by drawing the right triangles corresponding to the angles involved.
sin2(x) + cos2(x) = 1
sin(-x) = -sin(x)
cos(-x) =  cos(x)
sin(x)  = sqrt(1-cos2(x))
cos(x)  = sqrt(1-sin2(x)) 

sin(a+90°) =  cos(a)
sin(a-90°) = -cos(a)
cos(a+90°) = -sin(a)
cos(a-90°) =  sin(a)

sin(90°-a) =  cos(a)
cos(90°-a) =  sin(a)

tan(a)  = sin(a)/cos(a)
ctan(a) = 1/tan(a) = cos(a)/sin(a)

Special Equations

Sum of angles

sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
sin(a-b) = sin(a)*cos(b) - cos(a)*sin(b)
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)
cos(a-b) = cos(a)*cos(b) + sin(a)*sin(b)
tan(a+b) = [tan(a)+tan(b)]/[1-tan(a)*tan(b)]

Sum of sine and cosine

sin(a) + sin(b) =  2 * sin((a+b)/2) * cos((a-b)/2)
sin(a) - sin(b) =  2 * cos((a+b)/2) * sin((a-b)/2)
cos(a) + cos(b) =  2 * cos((a+b)/2) * cos((a-b)/2)
cos(a) - cos(b) = -2 * sin((a+b)/2) * sin((a-b)/2)

Product of sine and cosine

sin(a) * cos(b) = [sin(a+b) + sin(a-b)]/2
cos(a) * cos(b) = [cos(a+b) + cos(a-b)]/2
sin(a) * sin(b) = [cos(a-b) - cos(a+b)]/2


sin(a/2) = ±sqrt( (1-cos(a)) / 2 )
cos(a/2) = ±sqrt( (1+cos(a)) / 2 )

tan(a/2) = ±sqrt( (1-cos(a)) / (1+cos(a)) ) 
         = sin(a) / (1+cos(a))
         = (1-cos(a)) / sin(a)
In the equations above, the correct sign of the sqrt() result must be chosen from the specific configuration.

Reverse Identities

asin(-x) = -asin(x)
acos(-x) =  acos(x)

acos(x) + acos(-x) = 180° (π)
asin(x) + acos(x) = 90° (π/2)

Graphical Interpretation of Trigonometric Relations

The reverse functions "acos(x)" and "asin(x)"

The inverse functions "1/cos(x)" and "1/sin(x)"

The products of "cos(x)" and "sin(x)" values

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