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}