Trigonometric Identities & Equations Vocab

triangle area formula

Area = (1/2) bc sin A b = base c = length of short side A = angle between

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, for any triangle ABC: a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B c² = a² + b² - 2ab cos C

vector

a quantity that has both magnitude and direction

displacement vectors

a vector that represents distance and direction

odd function

always has opposite values for an input and its opposite, it is symmetric about the origin f(-x) = -f(x)

even function

always has the same value for an input and its opposite, it is symmetric with respect to the y-axis f(-x) = f(x)

trigonometric equation

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

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

Even Identities

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

solve a triangle

determine the measures of all of the angles and sides of a triangle

Cofunction Identities

for x measured in radians (including reciprocal functions), replace 90° with π/2 for x measured in degrees: sin x = cos (90° - x) cos x = sin(90° - x) tan x = cot (90° - x) cot x = tan (90° - x) sec x = csc(90° - x) csc x = sec(90° - x)

Phase Identities

for x measured in radians, replace 90° with π/2 for x measured in degrees: cos (x - 90°) = sin x sin(x + 90°) = cos x sin (x + 180°) = -sin x cos(x + 180°) = -cos x

Reciprocal Identities

identities based on the definitions of the trigonometric functions Sine & cosecant: csc x = 1/sin x, sin x = 1/csc x sin x csc x = 1 Cosine & secant: sec x = 1/cos x, cos x = 1/sec x cos x sec x = 1 Tangent & cotangent: cot x = 1/tan x, tan x = 1/cot x tan x cot x = 1 Tangent, cotangent: sine & cosine tan x = sin x/cos x, cot x = cos x/sin x

ambiguous case

more than one possible answer, if known angle is acute (in SSA) there may be 1/2/no triangles that match the measures given

scalar quantities

numbers that state magnitude but do not have direction, as opposed to vectors

Law of Sines

proportions that hold for the sines of the angles of a triangle and the sides opposite the angles sin A/a = sin B/b = sin C/c a/sin A = b/sin B = c/sin C

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

SSA

side-side-angle, the law of sines can be used to solve a triangle when two sides and an angle not between them are known (if known angle is acute, 2 solutions are possible)

SSS

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

Double-Angle Identities

sin 2a = 2 sin a cos a cos 2a = cos²a - sin²a cos 2a = 1 - 2 sin²a cos 2a = 2 cos²a - 1

Half-Angle Identities

sin a/2 = ±√((1-cos a)/2) cos a/2 = ±√((1+cos a)/2)

Odd Identities

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

Sum and Difference Identities

sin(A+B) = sin A cos B + cos A sin B sin(A−B) = sin A cos B − cos A sin B cos(A−B) = cos A cos B + sin A sin B cos(A+B) = cos A cos B − sin A sin B

primary solutions

solutions to trigonometric equations that are between 0° and 360°, or between 0 and 2π radians

phase shift

the horizontal shifting of a periodic function

Pythagorean Identities

these are trigonometric identities that are derived using the Pythagorean theorem, can be derived using the reciprocal triangle functions sin² x + cos² x = 1 1 + tan² x = sec² x cot² x + 1 = csc² x

oblique triangles

triangles that are not right triangles