Trigonometrics!

Law of Cosines

In the case of SSS, SAS, the law of sines does not work a^2 = b^2 + c^2 - 2bc(cosA) b^2 = a^2 + c^2 - 2ac(cosB) c^2 = b^2 + c^2 - 2bc(cosC)

shortcuts for multiplication and division

Let z1 and z2 be complex numbers where z1 = r1 (cos θ1 + i sin θ1) z2 = r2 (cosθ2 + i sin θ2) then: z1z2 = r1r2(cos (θ1 + θ2) + i sin (θ1 + θ2)) z1/z2 = r1/r2(cos (θ1 + θ2) + i sin(θ1 - θ2))

SOHCAHTOA

The mnemonic phrase used for the equations: sinx = opposite/hypotenuse cosx = adjacent/hypotenuse tanx = opposite/hypotenuse to find the sides and angles of a right triangle, when two others are present

addition of vectors (numerically)

add components

Coterminal Angles

angles that share the same initial side and terminal sides. can be found by adding or subtracting pi or 360 degrees to the angle.

Radians

another unit of measurement, describing the amount of distance the unit is around the unit circle to convert radians to degrees: pi/180 = x

Six trig functions of a point not on the unit circle

cosθ = x/r sinθ = y/r tanθ = y/x secθ = r/x cscθ = r/y cotθ = x/y when x^2 + y^2 = r^2

Vectors

directed line segment that shows both magnitude and direction

direction angles

if u is a unit vector such that theta is the angle from the positive x-axis to u, the terminal point of u lies on the unit circle, and; u = <costheta, sintheta> = (costheta)i + (sintheta)j θ is called the direction angle of the vector.

addition of vectors (pictorially)

move vectors so initial points of one meets the terminal point of another

Law of Sines

must be one of the following: AAS, ASA, SSA a/sinA = b/sinB = c/sinC

Demoivre's Theorem

raising a complex number to a power z^n = r^n(cos(nθ) + i sin(nθ))

Examples of how to verify

sec^2 x - 1/sec^2 x = sin^2 x tan^2 x/sec^2 x sin^2 x/cos^2 x / 1/cos^2 x sin^2 x / 1 1/(1-sin^2 x) + 2/(1+sin^2 x) = 2sec^2 x ((1+sin^2 x) + (1-sin^2 x)) / (1-sin^2 x)(1+sin^2 x) 2/(1-sin^2 x) 2/cos^2 x 2sec^2 x (tan^2 x + 1)(cos^2 x - 1) = -tan^2 x (sec^2 x)(-sin^2 x) (1/cos^2 x)(-sin^2 x / 1) -sin^2 x/cos^2 x -tan^2 x

Half-angle formula

sin x/2 = +/- sqrt ((1-cosx)/2) cos x/2 = +/- sqrt ((1 + cosx)/2) tan x/2 = (1 - cosx)/sinx = sinx/(1-cosx)

Sum and difference formulas

sin(u + v) = (sinu)(cosv) + (cosu)(sinv) sin(u - v) = (sinu)(cosv) - (cosu)(sinv) cos(u + v) = (cosu)(cosv) - (sinu)(sinv) cos(u - v) = (cosu)(cosv) + (sinu)(sinv) tan(u + v) = (tanu + tanv) / (1 - (tanu)(tanv)) tan(u - v) = (tanu - tanv) / (1 - (tanu)(tanv))

Double angle formulas

sin2u = 2(sinu)(cosu) cos2u = cos^2 u - sin^2 u = 2cos^2 u - 1 = 1 - 2sin^2 u tan2u = (2tanu)/(1 - tan^2 u)

The Trig Functions

sinx cosx tanx cscx secx cotx

Fundamental trig identities

sinx = 1/cscx cosx = 1/secx tanx = 1/cotx (cotx)(tanx) = 1 cscx = 1/sinx secx = 1/cosx cot = 1/tanx tanx= sinx/cosx cotx = cosx/sinx (sinx)(cotx) = cosx

Pythagorean identities

sinx^2 x + cos^2 x = 1 1 + tan^2 x = sec^2 x cos^2 x = 1 - sin^2 x 1 + cot^2 x = csc^2 x

Angular speed

the scalar measure of rotation rate 2pi/t angular speed = V/r where V = linear speed, r = radius of circle

Linear speed

the speed in which an object moves in a linear path v = s/t where s = the distance traveled, and t = the time elapsed

Angle between two vectors

to find the angle, 0 ≤ θ ≤ 180 between the two vectors (neither are zero vectors) uv/||u||(||v||) OR uv = ||u||||v||cosθ

Component form of a vector

v = <v1, v2>

unit vector

vector where length = 1 v/||v||

How to use a calculator to evaluate functions

y = asin(bx+c) + d y=acos(bx+c) + d a:amplitude (height) b:period(wavelength) *how many waves occur in 2pi c:horizontal translation +c:to left -c:to right d:vertical translation

Dot product

yields a scalar and not a vector u = <v1, v2> v = <v1, v2> uv = (u1)(v1) + (u2)(v2)

Trig form of a complex number

z = a + bi = (r cosθ) + (r sinθ) i

Absolute value of a complex number

|a + bi| = sqrt: (a^2 + b^2)

Rules for verifying

1) work only with one side of the equation. 2) look for factoring, adding fractions, square binomials or common denominators. 3) look for ways to substitute identities. 4) if all else fails, convert everything to sines/cosines. 5) try something!

Arc length

2pi r(θ/360) = arc length

Examples of how to solve equations using identities

2sinx - 1 = 0 2sinx = 1 sinx = 1/2 3tan^2 x - 1 = 0 3tan^2 x = 1 tan^2 x = 1/3 tanx = sqrt (1/3)

Degrees

360 degrees in a circle to convert degrees to radians: 180/pi = x