Precalculus Study Guide: Triangles, Polar Coordinates, Vectors, Complex Numbers, and Conic Sections

Solving Triangles

Right Triangles and Inverse Trigonometric Functions

Right triangles are triangles with one 90-degree angle. Solving right triangles involves finding unknown side lengths or angle measures using trigonometric ratios and their inverses.

• Trigonometric Ratios: Sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides.

• Inverse Trigonometric Functions: Used to find an angle when two sides are known.

• Formulas:

  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

  • $\theta = \arcsin\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$ (and similarly for $\arccos$ and $\arctan$)

• Example: Given a right triangle with legs 3 and 4, find the angles.

Oblique Triangles and the Laws of Sines and Cosines

Oblique triangles have no right angle. The Law of Sines and Law of Cosines are used to solve for unknown sides or angles.

• Law of Sines: Used for ASA, SAA, and SSA cases.

• Law of Cosines: Used for SSS and SAS cases.

• Formulas:

  • $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ (Law of Sines)

  • $c^2 = a^2 + b^2 - 2ab\cos C$ (Law of Cosines)

• Inverse Trigonometric Functions: Used to find unknown angles in oblique triangles.

• Example: Given $a=7$, $b=10$, $C=45^\circ$, use Law of Cosines to find $c$.

Polar and Rectangular Coordinates

Coordinate Systems and Conversions

Points in the plane can be represented in rectangular (Cartesian) or polar coordinates. Converting between these systems is essential for analyzing curves and equations.

• Rectangular Coordinates: $(x, y)$

• Polar Coordinates: $(r, \theta)$, where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis.

• Conversion Formulas:

  • $x = r\cos \theta$

  • $y = r\sin \theta$

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \arctan\left(\frac{y}{x}\right)$

• Transforming Equations: Equations can be rewritten from rectangular to polar form and vice versa using the above relationships.

• Example: Convert $(3, 3)$ to polar coordinates.

Applications of Triangle Principles

Modeling and Solving Applied Problems

Trigonometric principles are used to solve real-world problems involving distances, heights, and navigation.

• Right Triangle Applications: Height of a building, distance across a river, etc.

• Law of Sines and Cosines Applications: Surveying, navigation, physics problems.

• Example: Find the height of a tree given the distance from the base and the angle of elevation.

Technology in Trigonometry

Calculators and software are used to approximate values and solve triangles efficiently.

• Calculator Functions: Trigonometric and inverse trigonometric functions, solving for sides and angles.

• Example: Use a calculator to find $\sin^{-1}(0.5)$.

Vectors

Vector Representations and Operations

Vectors are quantities with both magnitude and direction, represented in various forms and manipulated using algebraic operations.

• Representations: Component form $(a, b)$, magnitude and direction, unit vectors.

• Finding a Unit Vector: Divide a vector by its magnitude.

• Magnitude: $|\vec{v}| = \sqrt{a^2 + b^2}$

• Direction: $\theta = \arctan\left(\frac{b}{a}\right)$

• Vector Addition/Subtraction: Add/subtract corresponding components.

• Scalar Multiplication: Multiply each component by the scalar.

• Dot Product: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$

• Angle Between Vectors: $\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|\,|\vec{b}|}$

• Parallel Vectors: One is a scalar multiple of the other.

• Orthogonal Vectors: Dot product is zero.

• Example: Find the unit vector in the direction of $(3, 4)$.

Complex Numbers in Rectangular and Polar Forms

Definitions and Operations

Complex numbers can be represented in rectangular form $(a + bi)$ or polar form $(r(\cos \theta + i\sin \theta))$.

• Modulus: $r = \sqrt{a^2 + b^2}$

• Argument: $\theta = \arctan\left(\frac{b}{a}\right)$

• Multiplication/Division in Polar Form:

  • Multiplication: $r_1r_2[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)]$

  • Division: $\frac{r_1}{r_2}[\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)]$

• Conversion: Use $a = r\cos \theta$, $b = r\sin \theta$ to go from polar to rectangular, and the modulus/argument formulas to go from rectangular to polar.

• Example: Convert $1 + i$ to polar form.

Applications of Vectors

Modeling with Vectors

Vectors are used to model physical quantities such as force, velocity, and displacement in applied problems.

• Example: Find the resultant force given two forces represented as vectors.

Conic Sections

Standard Equations and Features

Conic sections include circles, parabolas, ellipses, and hyperbolas, each with a standard equation and defining features.

• Completing the Square: Used to rewrite general equations into standard form.

• Classification: Determined by the coefficients in the general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

• Circle: $(x-h)^2 + (y-k)^2 = r^2$

• Parabola: $(y-k)^2 = 4p(x-h)$ or $(x-h)^2 = 4p(y-k)$

• Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

• Key Features:

  • Circle: Center $(h, k)$, radius $r$

  • Parabola: Vertex, focus, directrix

  • Ellipse: Center, vertices, foci

  • Hyperbola: Center, vertices, foci, asymptotes

• Example: Identify the type and features of $x^2 + y^2 - 4x + 6y + 9 = 0$.

Graphing Conic Sections

Graphing involves plotting the key features and using the standard form equations.

• Circle: Plot center and radius.

• Parabola: Plot vertex, focus, and directrix.

• Ellipse: Plot center, vertices, and foci.

• Hyperbola: Plot center, vertices, foci, and asymptotes.

• Classification: Use the standard or general form to determine the type.

• Example: Graph $(x-2)^2 + (y+1)^2 = 9$.

Applications of Conic Sections

Conic sections model real-world phenomena such as planetary orbits (ellipses), satellite dishes (parabolas), and navigation systems (hyperbolas).

• Example: Use the equation of a parabola to model the path of a projectile.

Conic Section	Standard Equation	Key Features
Circle	$(x-h)^2 + (y-k)^2 = r^2$	Center $(h, k)$, radius $r$
Parabola	$(y-k)^2 = 4p(x-h)$	Vertex $(h, k)$, focus, directrix
Ellipse	$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$	Center $(h, k)$, vertices, foci
Hyperbola	$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$	Center $(h, k)$, vertices, foci, asymptotes

Additional info: Some examples and context were inferred to ensure completeness and clarity for Precalculus students.