Math ch.8
Remember for some general/principle solutions problems...
Add the k•360 degrees or k•180 degrees depending on how big the angle is already
When it says find the principle solution and it gives you 3sec^2x = 4...
Change the sec to cosine: 1/cos^2x = 4/3 1/cosx = +- square root of 4/3 Cosx = +- square root of 3/4 or (square root of 3)/2 Since it is +- there are solutions in every quadrant(4solutions)
Double angle formulas for cosine
Cos2x = 2cos^2-1 Cos2x = 1-2sin^2x Cos2x = cos^2x - sin^2x
State whether the result is even, odd, or neither
Ex) the product of two odd functions If f and g are both odd, then f•g(-x) = [f(-x)•g(-x)] = [(-f(x)•(-g(x))] = [f(x)•g(x)] = f•g(x) Therefore the product of 2 odd functions is an even function
Lead and lag is when
Ex) y=cos(2x+pi/2)=cos2(x+pi/4) and this means lags to the left pi/4 units Ex) y=3cos(pi/2x-pi)=3cospi/2(x-2) and this means leads to the right 2 units
Find cot(tan^-1(3)-tan^-1(2))
Let tan^-1(3) = x and tan^-1(2) = y tan x = 3 and tan y = 2 tan(x-y) = (tanx-tany)/1+tanxtany = 1/7 cot(tan^-1(3)-tan^-1(2)) = 7
Find sin^-1(-0.5)
Let y = sin^-1(-0.5) Then sin y = -0.5 and -pi/2<=y<=pi/2 Therefore y = 11pi/6 and sin^-1(-0.5) = 11pi/6 Do the same for cos^-1 but with range 0<=y<=pi
Product-to-sum formulas
Sina(sinb) = 0.5(cos(a-b) - cos(a+b)) Cosa(cosb) = 0.5(cos(a-b) + cos(a+b)) Sina(cosb) = 0.5(sin(a-b) + sin(a+b))
Sum-to-product formulas
Sins+sint = 2sin((s+t)/2)cos((s-t)/2) Sins-sint = 2cos((s+t)/2)sin((s-t)/2) Coss+cost = 2cos((s+t)/2)cos((s-t)/2) Coss-cost = -2sin((s+t)/2)sin((s-t)/2)
Periodic function is odd if...
The graph is symmetric with respect to the origin
Periodic function is even if...
The graph is symmetric with respect to the y-axis(reflect across y-axis and be symmetrical)
When they ask what the period of f(0.5x) is given f(x) is 6
The period is 12 bc it is the period divided by the number multiplied by x. Any translation up, down, left, right or number in front of f(x) like amplitude does not affect the period
Theorem to solve the given equation
acosx + bsinx = c cos(x-y) where c = square root of a^2 + b^2 and cos y = a/c and sin y = b/c
Pythagorean identities
sin^2x + cos^2x = 1 tan^2x + 1=sec^2x 1+cot^2x = csc^2x
Inverse sec and csc domain
|x|>=1 and range is all numbers
Half formula for cosine
Cos(x/2) = +- square root of (1+cosx)/2 Sign is determined by the quadrant of x/2
Prove the statement
Cos^-1(3/5) - sin^-1(3/5) = sin^-1(7/25) Let cos^-1(3/5) = x and sin^-1(3/5)=y cos x=3/5 Sin y = 3/5 sin x=4/5. Cos y = 4/5 x - y = sin^-1(7/25) sin(x-y) = sinxcosy-cosxsiny = (4/5)(4/5)-(3/5)(3/5) = 7/25
Solve the equation given answers to the nearest 0.1 degrees or 0.01 radian
Ex) 3cosx + 4sinx = 2 a=3, b=4, c= square root of a^2 + b^2 Find cos beta = 3/5 and sin beta = 4/5 Beta = 53.13 degrees 3cosx+4sinx = 5cos(theta - 53.13 degrees) = 2 5cos(theta - 53.13) = 2 Solve for theta
When it says graph both curves and solve a trigonometric equation to find the coordinates of their points of intersection, then...
Ex) given y=cosx and y=sin2x Cosx=sin2x Cosx = 2sinxcosx 2sinxcosx-cosx = 0 Cosx(2sinx-1) = 0 X = 90,270 or x=30,150 So... x=90 + 180k or x = 30+360k, 150+360k
Express w/o using trigonometric or inverse trig functions.
Ex)cos(sin^-1x) Let y=sin^-1x. Then cos(sin^-1x) = cosy, where sin y = x and -pi/2<=y<=pi/2. Cos(sin^-1x) = cosy = positive square root of 1-sin^2y = square root of 1-x^2
Addition formulas for sine and cosine
Sin(s+t) = sinscost + cosssint Sin(s-t) = sinscost - cosssint Cos(s+t) = cosscost - sinssint Cos(s-t) = cosscost + sinssint
Half-angle formula for sine
Sin(x/2) = +- square root of (1-cosx)/2 Sign is determined by the quadrant of x/2
Double angle formulas for sine
Sin2x = 2sinxcosx
Find general solutions of sin3x = cos3x
Sin3x = cos3x Tan3x = 1 3x = 45, 225 3X = 45 + 180k X = 15 + 60k X = 15,75,135,195,255,315 all the way up until u reach 360 but don't go over U add 180k and not 360k because the difference between 225-45 = 180
Addition/subtraction formulas for tangent
Tan(s+t) = (tans + tant)/1-tanstant Tan(s-t) = (tans - tant)/ 1+tanstant
Half angle formulas for tangent
Tan(x/2) = (1-cosx)/sinx Tan(x/2) = sinx/(1+cosx)
Double angle formula for tangent
Tan2x = (2tanx)/1-tan^2x
Reject solutions like sinx = -2 because
The margin for sin/cos is -1<= x <= 1
Depending on what quadrant the sin/cos value should be
Use reference angle (180-x or 360-x) to find those angles
When it says express the product as a sum
Use the product-to-sum formulas
When it says to express the difference as a product
Use the sum-to-product formulas
For graphing y=3cosx - 4sinx
Use y=5cos(theta - (-53.13))
Inverse cosine domain and range
{x:-1<=x<=1} and {y:0<=y<=pi}
Inverse tangent domain and range
{x:-infinity<x<infinity} and {y:-pi/2<y<pi/2}
Inverse cotangent domain and range
{x:-infinity<x<infinity} and {y:0<y<pi}
Inverse sine domain and range
{x:-pi/2<=x<=pi/2} and {y:-pi/2<=y<=pi/2}