Naming angles

Building an angle

Each side of an angle is bounded by a ray (a directed line segment). When we put two rays together with a common endpoint, the endpoint is called the vertex of the angle, and the two rays are the sides of the angle.

There are different systems to measure angles. Typically, degrees and radians are used. 360360 degrees is equivalent to 2π2\pi radians.

Greek letters are typically used as variables for the measure of an angle. Theta θ\theta is highly common.

Types of angles

Angle in degrees Angle in radians Angle name
θ=0°\theta = 0\degree θ=0\theta = 0 Zero angle
0°<θ<90°0\degree < \theta < 90\degree 0<θ<π20 < \theta < \frac{\pi}{2} Acute angle
θ=90°\theta = 90\degree θ=π2\theta = \frac{\pi}{2} Right angle
90°<θ<180°90\degree < \theta < 180\degree π2<θ<π\frac{\pi}{2} < \theta < \pi Obtuse angle
θ=180°\theta = 180\degree θ=π\theta = \pi Straight angle
180°<θ<360°180\degree < \theta < 360\degree π<θ<2π\pi < \theta < 2\pi Reflex angle
θ=360°\theta = 360\degree θ=2π\theta = 2\pi Complete angle

Complimentary and supplementary angles

Complimentary angles are two angles that sum to 90°90\degree or π2\frac{\pi}{2}.

Supplementary angles are two angles that sum to 180°180\degree or π\pi.

Trigonometric identities

Basic identities

  • tanθ=sinθcosθ=ln(cosθ)\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \ln(\cos{\theta})
  • cotθ=cosθsinθ=ln(sinθ)\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \ln(\sin{\theta})

Pythagorean identities

The fundamental Pythagorean Trigonometric identity is:

sinx+cosx=1\sin x + \cos x = 1

From this formula we can derive the formulas for other functions:

Sine and cosine:

sin2x+cos2x=1\sin^2 x + \cos^2 x = 1
  • cos2x=1sin2x\cos^2 x = 1 - \sin^2 x
  • sin2x=1cos2x\sin^2 x = 1 - \cos^2 x

Tangent and secant:

1+tan2x=sec2x1 + \tan^2 x = \sec^2 x
  • tan2x=sec2x1\tan^2 x = \sec^2 x - 1
  • 1=sec2xtan2x1 = \sec^2 x - \tan^2 x

Cotangent and cosecant:

1+cot2x=csc2x1 + \cot^2 x = \csc^2 x
  • cot2x=csc2x1\cot^2 x = \csc^2 x - 1
  • 1=csc2xcot2x1 = \csc^2 x - \cot^2 x