College Algebra, Pre-Calculus, & Trigonometry Explained

1. Determine if two angles are coterminal: Coterminal angles are two angles whose terminus (point where the ray emanates from) lie on the same arc of a circle. To determine if two angles are coterminal, you need to know or find their measures and check if they have the same measure or differ by less than 180 degrees (a full circle).

2. Finding angle measures for similar triangles: If you have one triangle and another triangle that is similar to it, the ratio of their corresponding sides will be the same as the ratio of their corresponding angles. You can use this fact to find unknown angle measures in similar triangles.

3. Finding side lengths for similar triangles: By using the ratios from the properties of similar triangles (AA criterion, SAH criterion), you can find the missing side lengths by setting up proportions based on known sides and angles.

4. The Law of Sines (ASA, SAA, and SSS): These are different cases of the Law of Sines. ASA stands for "Angle-Side-Angle," SAA for "Side-Angle-Angle," and SSS for "Side-Side-Side." Each case allows you to solve for a missing angle or side length in a triangle.

5. The Law of Cosines (SAS, SSS): The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used when two sides and their included angle are known, or when all three sides are known.

6. The Area of a Triangle using Heron's Formula: Heron's Formula allows you to calculate the area of a triangle when you know the lengths of all three sides.

7. Finding the Component Form of a Vector: This involves breaking down the vector into its horizontal (x-component) and vertical (y-component).

8. Finding the Magnitude of a Vector: The magnitude (or length) of a vector is the square root of the sum of the squares of its components.

9. Multiplying a Vector by a Scalar: This is done by multiplying each component of the vector by the scalar value.

10. How to Find the Unit Vector: A unit vector is a vector with a magnitude of 1. To find it, you divide a given vector by its magnitude.

11. How to Write a Linear Combination of Unit Vectors: This involves multiplying each unit vector in a given direction by the scalar that represents that component's contribution to the total vector.

12. Finding the Dot Product: The dot product (or scalar product) of two vectors is the product of their magnitudes and the cosine of the angle between them.

13. How to Find the Angle Between Two Vectors: You can use the dot product and the magnitudes of the two vectors to find this angle.

14. How to Determine if Two Vectors are Orthogonal: Two vectors are orthogonal (perpendicular) if their dot product is zero.

15. How to Add Two Complex Numbers Graphically: This involves placing each complex number on the complex plane and connecting them with a segment, then finding the midpoint of that segment as the resultant complex number.

16. Find the Trigonometric (Polar) Form of a Complex Number: This involves expressing a complex number in terms of its magnitude (r) and angle (θ), where r is the distance from the origin to the point, and θ is the angle between the positive x-axis and the line segment connecting the origin to the point.

17. Convert a Complex Number between Rectangular Form and Polar Form: This involves converting back and forth between the standard form (a + bi) and the polar form (r(cos(θ) + i sin(θ))).

18. Multiplying Complex Numbers in Polar Form: To multiply two complex numbers in polar form, multiply their magnitudes and add their angles.

19. Dividing Complex Numbers in Polar Form: To divide two complex numbers in polar form, divide their magnitudes and subtract their angles.

20. Product and Quotient Theorems: These theorems relate the trigonometric functions of an angle to the next higher and lower positive angles (Product Theorem) or to the angles obtained by adding or subtracting π from the original angle (Quotient Theorem).

21. De Moivre's Theorem: This theorem states that for any complex number in polar form, its N-th power is also a complex number in polar form whose magnitude is the N-th power of the original magnitude and whose angle is N times the original angle, divided by 360° (or 2π radians).

22. Powers of Complex Numbers in Polar Form: This involves raising a complex number in polar form to a given power using De Moivre's Theorem.

23. Roots of Complex Numbers in Polar Form: To find the N-th roots of a complex number, you divide the angle by N and take the N-th root of the magnitude.

24. Solving Equations Using Roots of Complex Numbers: This involves finding all solutions to an equation whose complex numbers are expressed in polar form.

25. Convert between Degrees and Radians: Since some trigonometric identities, like De Moivre's Theorem, involve radian measure, it's important to be able to convert back and forth between degrees and radians.

26. Finding the Area of a Polygon: The Law of Cosines can also be used to find the area of a polygon when you know the lengths of its sides and the measures of its interior angles.

27. How to Solve Systems of Linear Equations: Various methods such as graphing, substitution, elimination, matrices, or Cramer's Rule can be used to find the solutions to a system of linear equations.

28. Finding the Normal to a Curve: The normal (perpendicular) to a curve at a given point is the vector perpendicular to the tangent vector at that point.

29. Work-Energy Theorem: This theorem relates the work done on an object to its change in kinetic energy.

30. Conservation of Energy and Momentum: In closed systems, total mechanical energy (kinetic + potential) remains constant, and momentum is conserved if no external forces are acting upon the system.

These mathematical concepts and methods can be applied in various fields, including physics, engineering, computer science, economics, and more.