Math Semester Notes

Angles of Elevation

Angles of Elevation

Angle above the horizontal when looking up, used for calculating heights of objects.

Angles of Depression

Angle below the horizontal when looking down, used for calculating depths of objects.

Trigonometric Ratios - Soh Cah Toa:

sine

Opposite/Hypotenuse.

cosine

Adjacent/Hypotenuse.

tangent

Opposite/Adjacent.

Sine Law

Used for non-right-angled triangles, relates side lengths to sines of opposite angles (a/sin(A) = b/sin(B) = c/sin(C)).

Cosine Law

Used for non-right-angled triangles, relates side lengths to cosine of included angle (a² = b² + c² - 2bc * cos(A), b² = a² + c² - 2ac * cos(B), c² = a² + b² - 2ab * cos(C)).

Perpendicular Bisector

Divides a segment into two equal parts, creates right angles, intersects at the midpoint.

Altitude

Perpendicular segment from vertex to opposite side, can be inside, outside, or on the triangle.

Median

Segment from vertex to midpoint of opposite side, divides triangle into two equal areas.

Angle Bisector

Divides angle into two equal angles, intersects at the incenter.

Properties Acronym

"An Odd Peanut Butter Cup May Contain Apple Butter Instead" for Angle Bisector, Perpendicular Bisector, Circumcenter, Median, Centroid, Altitude, Incenter.

Triangle Midsegment Theorem

Midsegment is half the length of the parallel side.

Square

All sides and angles are equal, diagonals bisect each other at right angles.

Rectangle

Opposite sides are equal and parallel, all angles are right angles.

Parallelogram

Opposite sides are equal and parallel, opposite angles are equal.

Rhombus

All sides are equal, opposite angles are equal.

Trapezoid

One pair of opposite sides are parallel, non-parallel sides have different lengths.

Area Triangles

½ Bh

Area Parallelograms

Bh

Area Trapezoids

½ (b1+b2) h

Area Rhombus/Kites

½ (dd)

Area Circles

pi r squared

Area Polygons

½ ap

LSA & TSA Prisms

Lateral Surface Area = ph Total Surface Area = ph +2b

LSA & TSA Cylinders

Lateral Surface Area = 2πrh, Total Surface Area = 2πrh + 2 pi r squared

LSA & TSA Pyramids

Lateral Surface Area = ½ pl, Total Surface Area = ½ pl + base

LSA & TSA Cones

Lateral Surface Area = πrl, Total Surface Area = πrl +πr²

Spheres

Total Surface Area = 4πr², Volume = 1/2(3πr²)

Arc Length Formula

arc/360 = angle in radians/π x diameter

Sector Area Formula

arc/360 = angle in radians/π x radius²

Segment Area Formula

Sector area - Triangle area

Properties of Radii and Chords

Equal chords have equal arcs

Properties of Tangents

Tangents from a point are equal in length

Volume Cylinder:

V = πr²h

Volume Prism:

V = bh

Volume Pyramid:

V = 1/3 bh

Volume Cone:

V = 1/3πr²h

Volume Sphere:

V = 4/3πr³

Volume Sphere:

V = 1/2(3πr²)