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