Precalc Unit 3B

Tangent asymptote equation

π/2b + k(π/b), where k is any integer

Tangent period

π/b

Inverse sin graph (sin^-1(x))

Restricted domain: [-1,1]

Range: [-π/2, π/2]

Inverse cos graph (cos^-1(x))

Restricted Domain: [0, π]

Range: [-1,1]

Inverse tan graph (tan^-1(x))

Restricted domain: (-π/2,π/2)

Range: idk dude

Find inverse of the function

Swap x and y, solve. Domain and range change depending on vertical/horizontal shifts/dilations

Vertical dilations change…

DISTANCE from x-axis. If a point is on the x-axis it doesn’t move.

Solving trigonometric equations (RESTRICTED DOMAIN)

answer is negative: apply reference angle to quadrants where sin, cos, or tan of the angle is negative.

Answer is positive: apply reference angle to quadrants where sin, cos, or tan of the angle is positive.

Answer is BOTH (FACTORS AND/OR SQUARE ROOT!): Apply reference angle to ALL quadrants.

Solving trigonometric equations (ALL VALUES)

Answer will be an EQUATION. Otherwise same as restricted domain.

csc(x)

1/sin(x)

sec(x)

1/cos(x)

cot(x)

1/tan(x)

csc and sec graphs

Graph function normally, midline points become VAs and make the U graphs.

cot graph

tan graph reflected across y axis

1

cos²(x) + sin²(x)

cos²(x)

1 - sin²(x)

sin²(x)

1 - cos²(x)

csc²(x)

1 + cot²(x)

cot²(x)

csc²(x) - 1

sec²(x)

tan²(x) + 1

tan²(x)

sec²(x) - 1

Tips to rewrite in single trig identity

1. Rewrite in sin and cos

2. Squared trig functions could be a pythagorean identity

3. Put in terms of one trig function

sin(a + b)

sin(a)cos(b) + sin(b)cos(a)

sin(a - b)

sin(a)cos(b) - sin(b)cos(a)

cos(a + b)

cos(a)cos(b) - sin(a)sin(b)

cos(a - b)

cos(a)cos(b) + sin(a)sin(b)

sin(2a)

2sin(a)cos(a)

cos(2a)

cos²(a) - sin²(a)

1 - 2sin²(a)

2cos²(a) - 1

Polar —> Rectangular COORDINATES

x = rcos(x)

y = rsin(x)

(r, theta) —> (x,y)

Rectangular —> Polar COORDINATES

r = √x² + y²

theta = tan^-1(y/x)

PAY ATTENTION TO QUADRANTS ON THETA.

Name 4 diff. ways

r: 2 pos. and 2 neg.

theta accordingly

Polar Form

r[cos(x) + isin(x)]

rcos(x) + risin(x)

Rectangular complex numbers —> Polar FORM

r = √x² + y²

theta = tan^-1(y/x)

PAY ATTENTION TO QUADRANTS: regular complex number = a + bi, use negative a and b to determine quad.

Polar complex numbers —> Rectuangular FORM

x = rcos(x)

y = rsin(x)

end in a + bi

sin circles

Positive: Opens up

Negative: Opens down

cos circles

Positive: Opens right

Negative: Opens left

cos roses

Starts on pole

Sin roses

start above pole

Polar functions: n is ODD

n petals

cycle: [0, π]

Polar functions: n is EVEN

2n petals

cycle: [0, 2π]

Max distance from pole

= amplitude

Find endpoints

Plug endpoints in for theta in rcos(theta) and rsin(theta)

Limaçon

r = a +- bcos(theta)

Limaçon a = b

cardioid

Limaçon a > b

dimpled cardioid

Limaçon a < b

inner loop limaçon

r = theta where theta >= 0

Spiral

Distance from pole is INCREASING

r is positive and increasing

r is negative and decreasing

Distance from pole is DECREASING

r is positive and decreasing

r is negative and increasing

Use AROC to estimate

f(x) = y1 + AROC(x - x1)