Trigonometric Identities & Equations

AAS

"angle-angle-side." the law of sines can be used to solve a triangle when two angles and a side not between the angles are known.

ASA

"angle-side-angle." the law of sines can be used to solve a triangle when two angles and the side between them are known.

SAS

"side-angle-side." the law of cosines can be used to calculate the third side of an oblique triangle, given the other two sides and the angle between them.

SSS

"side-side-side." the law of cosines can be used to find the measures of the angles of a triangle when the three sides are known.

one-to-one function

A function that matches each output with one input

Law of Cosines

A property of trigonometry that helps to solve oblique triangles. The law states that for any triangle ABC, a² = b² + c² − 2bc cos A or b² = a² + c² - 2ac cos B or c² = a² + b² - 2ab cos C

vector

A quantity that has both magnitude and direction.

displacement vectors

A vector that represents distance and direction

trigonometric equation

An equation with a variable in place of the value of an angle.

solve a triangle

Determine the measures of all of the angles and sides of a triangle.

ambiguous case

In SSA if the angle (A) is acute, one/two/none solutions are possible, so this situation is called the __________

primary solution

Solutions to trigonometric equations that are between zero (0) and 360 degrees, or between zero and 2π radians

SSA

Stands for "side-side-angle." In trigonometry, the law of sines can be used to solve a triangle when two sides and an angle not between them are known. This situation is referred to as SSA. However, if the angle (A) is acute, two solutions are possible,so this situation is called "the ambiguous case."

Pythagorean Identities

an identity derived using the Pythagorean Theorem sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, 1 + cot² θ = csc² θ

triangle area formula

area = 1/2bh sin A

Phase Identities

cos (x-90°) = sin x sin (x+90°) = cos x sin (x+180°) = -sin x cos (x+180°) = -cos x

Even Identities

cos(-x) = cos x sec(-x) = sec x

odd function

function that are symmetric about the origin. a trigonometric function in which f(-x) = -f(x)

even function

function that are symmetric with respect to the y-axis. a trigonometric function in which f(-x) = f(x)

Double-Angle Identites

identities created by using the same input value twice in the sum identities sin 2u = 2 sin u cos u, cos 2u = cos² u - sin² u = 2 cos² u - 1 = 1 - 2 sin²u, tan 2u = (2 tan u) / (1- tan² u)

Sum and Difference Identites

identities that allow you to use your knowledge to calculate the value of the cosine or sine of other angles sin (u±v) = sin u cos v ± cos u sin v, cos (u±v) = cos u cos v ±↓ sin u sin v, tan (u ± v) = (tan u ± tan v) / (1 ±↓ tan u tan v)

scalar vectors

quantities that state "how much" or "how many" but do not have a direction

Half-Angle Identities

sin a/2 = ±√(1-cos a)/2, cos a/2 = ±√(1+cos a) /2, sign is determined by the quadrant a/2

Reciprocal Identities

sin x = θ/ csc x, sec x - θ/ cos x, tan x = θ/ cot x, csc x = θ / sin x, cos x = θ/ sec x, cot x = θ/ tan x,

Cofunction Identities

sin( π/2 - x) = cos x, csc( π/2 -x) = sec x, sec (π/2 -x) = csc x, cos (π/2 - x) = sin x, tan (π/2 - x) = cot x, cot (π/2 - x) = tan x

Odd Identities

sin(-x) = -sin x csc(-x) = -csc x tan(-x) = -tan x cot(-x) = -cot x

Law of Sines

states proportions that hold for the sines of the angles of a triangle and the sides opposite the angles

phase shift

the resulting horizontal shift from the phase

oblique triangles

triangles that are not right triangles