Trigonometric Identities and Equations

Vocabulary and terminology from Chapters 9 and 10 regarding Trigonometric Identities, Equations, and Applications.

Identities

Equations that enable us to simplify complicated expressions and are true for all values in the domain of the variable.

Pythagorean Identities

A set of equations involving trigonometric functions based on the properties of a right triangle, such as $\sin^2(\theta) + \text{cos}^2(\theta) = 1$, $1 + \text{tan}^2(\theta) = \text{sec}^2(\theta)$, and $1 + \text{cot}^2(\theta) = \text{csc}^2(\theta)$.

Even Function

A function in which $f(-x) = f(x)$ for all $x$ in the domain; the graph is symmetric about the y-axis, such as the cosine function where $\text{cos}(-\theta) = \text{cos}(\theta)$.

Odd Function

A function in which $f(-x) = -f(x)$ for all $x$ in the domain; the graph is symmetric about the origin, such as the sine function where $\text{sin}(-\theta) = -\text{sin}(\theta)$.

Reciprocal Identities

Set of equations relating trigonometric functions that are reciprocals of each other, e.g., $\text{sin}(\theta) = \frac{1}{\text{csc}(\theta)}$ and $\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}$.

Quotient Identities

Identities defining the relationship between certain functions: $\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}$ and $\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}$.

Cofunction Identities

Identities based on complementary angles that state the sine of an angle equals the cosine of its complement, such as \text{sin}(\theta) = \text{cos}(\frac{\text{\pi}}{2} - \theta).

Double-angle Formulas

Identities derived from the sum formulas for sine, cosine, and tangent in which the two angles are equal (\alpha = \text{\beta}).

Reduction Formulas

Identities derived from double-angle formulas used to reduce the power of an expression involving even powers of sine or cosine to the first power of cosine.

Half-angle Formulas

Identities derived from reduction formulas used when an angle is half the size of a special angle, preceded by a $\pm$ sign depending on the quadrant.

Oblique Triangle

Any triangle that is not a right triangle.

Law of Sines

A law used to solve oblique triangles stating that the ratio of an angle measurement to its opposite side is equal across the triangle: \frac{\text{sin}(\text{\alpha})}{a} = \frac{\text{sin}(\text{\beta})}{b} = \frac{\text{sin}(\text{\gamma})}{c}.

Ambiguous Case (SSA)

A situation in which the lengths of two sides and the measurement of an angle opposite one of those sides are known, potentially resulting in zero, one, or two possible triangles.

Law of Cosines

States that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice their product and the cosine of the included angle, such as a^2 = b^2 + c^2 - 2bc\text{cos}(\text{\alpha}).

Heron's Formula

A formula used to find the area of an oblique triangle when all three sides are known: \text{Area} = \text{\sqrt{s(s-a)(s-b)(s-c)}} where $s$ is the semi-perimeter.

Semi-perimeter

One-half of the perimeter of a triangle, calculated as $s = \frac{1}{2}(a + b + c)$.

Polar Axis

The positive x-axis of the coordinate plane when viewed in the polar system.

Pole

The origin of the coordinate plane in a polar coordinate system.

Polar Coordinates

A coordinate system where points are labeled (r, \text{\theta}), where $r$ is the radius from the pole and \text{\theta} is the angle measured in radians from the polar axis.

Cardioid

A heart-shaped polar curve produced by formulas such as r = a + a\text{cos}(\text{\theta}) or r = a + a\text{sin}(\text{\theta}).

Lima\u00e7on

A family of polar curves named for the French word for "snail," which can include a dimple (one-loop) or an inner loop.

Lemniscate

A polar curve resembling an infinity symbol $\infty$ or a figure 8, centered at the pole.

Rose Curve

A polar shape that produces petal-like graphics, with $n$ petals if $n$ is odd and $2n$ petals if $n$ is even.

Archimedes' Spiral

A polar curve defined by the formula r = a\text{\theta}, characterized by an ever-widening, spiraling path.

Modulus

The absolute value of a complex number, representing the distance from the origin to the point in the complex plane, defined as |z| = \text{\sqrt{a^2 + b^2}}.

Argument

The angle of direction \text{\theta} in the polar form of a complex number.

De Moivre's Theorem

A theorem used to find the power of a complex number in polar form, stating that z^n = r^n[\text{cos}(n\text{\theta}) + i\text{sin}(n\text{\theta})].

Parametric Equations

A set of equations where $x$ and $y$ are expressed as functions of a third variable, often time $t$, such as $x = f(t)$ and $y = g(t)$.

Parameter

An independent variable (often time) upon which both $x$ and $y$ depend as functions.

Orientation

The direction or path traced along a parametric curve in terms of increasing values of the parameter.

Projectile Motion

A type of motion modeled by parametric equations where an object is propelled forward and upward, subject to gravity.