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Geometry and Trigonometry Rules
Last Update: August 26, 2025
Introduction
This post summarizes some useful geometric and trigonometric functions and rules.
Similar Triangles
Similar Triangle geometry: If two triangles are similar, then their corresponding angle measures are equal , and their corresponding side lengths are in the same ratio.
So for Similar Triangles ABC and DFE:
θ1=θ7 ; θ2=θ8 ; θ3=θ9
and
b/f = a/d = c/e
Exterior and Interior Angles of a Triangle
Exterior Angle Geometry: When a triangle’s side is extended, the resultant exterior angle formed is equal to the sum of the measures of the two opposite interior angles of the triangle:
So for Triangle ABC:
θ6 =θ1+ θ2
θ5= θ3 +θ1
θ4 =θ3 +θ2
Law of Cosines and Law of Sines
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C) ; Law of Sines
Picture: Law of Sines Triangle
Law of Cosines
a2= b2+c2 – 2bccos(A) ; Law of Cosines
b2= a2+c2 – 2accos(B) ; Law of Cosines
c2= a2+b2 – 2abcos(C) ; Law of Cosines
Picture: Law of Cosines Triangle
Right Triangle Trig Functions
For a right triangle as shown below:
Pythagorean Theorem: a2 + b2 = c2
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent
mnemonic: SohCahToa
Trigonometric Identities
From the Pythagorean theorem we know that
- a2 + b2 = c2
- sin(θ) = a/c
- cos(θ) = b/c
- tan(θ) = a/b
Divide Equation 1 by c2
a2/c2+ b2/c2= c2/c2
Simplify and substitute using 2. and 3. to get
sin2(θ) + cos2(θ) = 1 ; Pythagorean Identity
Divide equation 1 by a2
a2/a2+ b2/a2= c2/a2
Simplify and substitute in 4. and 2. to get
1+ 1/tan2(θ) = 1/sin2(θ)
or
1+ cot2(θ) = csc2(θ); Pythagorean Identity
Divide equation 1 by b2
a2/b2+ b2/b2= c2/b2
Simplify and substitute in 4. and 3. to get
tan2(θ)+ 1 = 1/cos2(θ)
or
1+ tan2(θ) = sec2(θ); Pythagorean Identity
These identities are always true.
For completeness, let’s prove that in the next section.
Derivation of Trig Identities from a Unit Circle
Consider a unit circle with a radius of 1 centered at the origin (0,0):
On a unit circle, by definition, every point on the circumference will have coordinates (a,b) = (x,y) = (cosθ,sinθ)
x = cosθ
y = sinθ
The equation for the unit circle is
x2 + y2 = 1
So , substituting for x and y we get
cos2θ + sin2θ = 1 ; Pythagorean Identity
Divide “cos2θ + sin2θ = 1″ by cos2θ to get
cos2θ /cos2θ+ sin2θ/cos2θ = 1/cos2θ
or
1 + tan2θ = sec2θ ; Pythagorean Identity
Divide “cos2θ + sin2θ = 1″ by sin2θ to get
cos2θ /sin2θ+ sin2θ/sin2θ = 1/sin2θ
or
cot2θ + 1 = csc2θ ; Pythagorean Identity
Radius of Curvature
Circle geometry: Consider the drawing below comparing the red shaded (flat) curve with the blue (steep) shaded curve.
If we fit a circle to each curve, we get the two circles shown in the drawing below.
Picture_Radius of Curvature of a Curve
The Radius of Curvature for each curve is the radius of the fitted circle.
- The Radius of Curvature of the red curve is longer. The red curve , relative to the blue curve, is described as being “flat”.
- The Radius of Curvature for the blue curve is shorter. The blue curve , relative to the red curve, is described as being “steep”.
If the curves are not perfectly spherical, the same concept is applied but the Radius of Curvature could vary from point to point on the surface.
With respect to lenses, the focal length (distance between the focal point and the optical center) of the lense will be proportional to the Radius of Curvature.
- A lens with A longer focal length will focus light rays to a point further away from the lens (and vice versa).
- For a spherical lens, the focal length is approximately half the radius of curvature of the lens surface.
- The above relationship holds true when the lens is thin compared to its radius of curvature.
- For non-spherical lenses, the relationship between focal length and Radius of Curvature is more complex.
Perpendicular Line to the Surface of a Circle
Circle geometry: Since we have a nice drawing of a circle above, we can also define the normal line or perpendicular line to the surface of a circle:
A line that is perpendicular (normal) to the surface of a circle (meaning a line that forms a right angle with the tangent line at a point on the circle) will always pass through the center of the circle.
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