ANALYTIC TRIG!!!!!!!

Trig Identities

•sin^2x + cos^2x = 1

tan^2x + 1 = sec^2x

1 + cot^2x = csc^2x

•sin(-x) = -sinx

cos(-x) = cosx

tan(-x) = -tanx

Proving Trig Identities

1. Start with one side

2. Use known identities

3. Convert to sines and cosines

Example:

Prove:

2tanxsecx = 1/(1-sinx) – 1/(1+sinx)

Solving Trig Equations

Example:

Tan^2x - 3 = 0

Double-Angle Formulas

Sine: sin2x = 2sinxcosx

Cosine: cos2x = cos^2x – sin^2x

Tangent: tan2x = (2tanx)/(1 – tan^2x)

Example:

If cosx = -2/3 and x is in quadrant II, find sin2x.

Half-Angle Formulas

Sinu/2 = +/- √(1-cosu)/2

Cosu/2 = +/- √(1+cosu)/2

Tanu/2 = (1-cosu)/sinu = sinu/(1+cosu)

Example:

Find the exact value of sin22.5°

VectorsComponent Form: y

v = <x2 – x1, y2 – y1>

Magnitude of a Vector: x

|v| = √a^2 + b^2

Algebraic Operations on Vectors

If u = <a1, b1> and v = <a2, b2>, then

u + v = <a1 + a2, b1+b2>

cu = <ca1, cb1>

Example:

If u = <2,-3> and v = <-1,2> find u + v and -3v.

u + v = <2 – 1, -3 + 2> = <1,-1>

-3v = <-3(-1), -3(2)> = <3, -6>

Vectors in terms of i and j

i = <1, 0> j = <0,1>

v = <a, b> = ai +bj

u = <5, -8>

u = 5i – 8j

Horizontal and Vertical Components of a Vector

Let v be a vector with magnitude |v| and direction θ.

v = <a,b> = ai + bj

a = vcosθ and b = vsinθ

Example:

An airplane heads due north at 300 mi/h. It experiences a 40 mi/h crosswind flowing in the direction N 30° E.

v

u

a) Express the velocity v of the airplane relative to the air, and the velocity u of the wind in component form.

v = 0i + 300j = 300j

u = (40cos60°)i + (40sin60°)j

u = 20i + 20√3j

u = 20i + 34.64j

b) Find the true velocity of the airplane as a vector.

w = u + v

w = (20i + 20√3j) + (300j)

w = 20i + (20√3 + 300)j

w = 20i + 334.64j

c) Find the true speed of the airplane.

Speed:

w = √20^2 + 334.64^2 = 335.2 mi/h

ADDITION AND SUBTRACTION

FORMULAS

SINE• Sin(s+t) = sinscost + cosssint

• Sin(s-t) = sinscost – cosssint

COSINE

• Cos(s+t) = cosscost – sinssint

• Cos(s-t) = cosscost + sinssint

TANGENT

• Tan(s+t) = (tans + tant)/(1 – tanstant)

• Tan(s-t) = (tans – tant)/(1 + tanstant)

INVERSE TRIGONOMETRIC

FUNCTIONS

Domain: [-1, 1]

Range: [-pi/2, pi/2]

Domain: [-1, 1]

Range: [0, pi]

Domain: all real numbers

Range: (-pi/2, pi/2)

z = r(cos + isin )

• … where r = modulus of z = √a2 + b2

COMPLEX NUMBERS… THE TRIGONMETRIC VARIETY

DOT PRODUCTS

u = (a1, b1) v = (a2, b2)

u • v = a1a2 + b1b2

DOT PRODUCT THEOREM

u • v = |u||v|cosθ