analytic trig!!!!!!!
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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
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