Math Analysis Midterm Review

slope formula

(y₂- y₁) / (x₂- x₁)

y= sinθ+4

+4 will move the whole period upward by 4 units. So, we need to add 4 to each of our y-values, now it is 3, 4, 3, 2, 3.

y= cotθ

- Asymptotes: θ = 0, θ = π - Period: π - Zero will be (π/2, 0) - Opposite direction as tan graph

Tan graph

- Period: π - Look like Graph of y = x^3

Compass circle

- REMEMBER always draw out the North and South lines

Csc Graph

- Reciprocal of Sin - Graph sinθ first, then find the Asymptotes and reflect it. Asymptotes are where y-values are ZERO! - REMEMBER Y VALUES, ARE 0, 1, 0, -1, 0

Sec Graph

- Reciprocal of cosθ - Graph cosθ first, then find the asymptotes, and reflect it. - REMEMBER Y VALUES, ARE 1, 0, -1, 0, 1

*ASS* cases

- one solution - two solutions - no solution

sin 11π/6

-1/2

csc 2π/3

-2

Inverse: Domains for sin y = arcsin

-π/2 ≤ y ≤ π/2 In Q1 +, Q4 -

Inverse: Domains for tan y = arctan

-π/2 ≤ y ≤ π/2 In Q1 +, Q4 -

arctan -√3

-π/3

arctan -1

-π/4

tan 11π/6

-√3/3

arctan 0

0

tan -π

0

Inverse: Domains for cos y = arccos

0 ≤ y ≤ π In Q1 +, Q2 -

sec 6π

1

Steps for Identities

1) Factor 2) Common Denom. 3) Complex Fraction 4) Conjugate of either the num. or denom. 5) Change all terms to sin/cos

cos -π/3

1/2

The Leaning Tower of Pisa in Italy leans at an angle of about 84.7 degrees. The figure shows that 171 feet from the base of the tower, the angle of elevation to the top is 50 degrees. Find the distance, to the nearest tenth of a foot, from the base to the top of the tower.

184.3 ft

If a certain store sold 150 TV sets in 1999, and sold 289 TV sets in 2002, how many TV sets can they expect to sell in 2004? In what year can they can expect to sell 750 TV sets?

1999 = 0, 2002 = 3 Find the difference between those two years and use two numbers that had the same difference *Time usually = x* (0, 150), (3, 289) - use the slope formula to find the slope, which is 46.3/1 - y = 46.3x + 150 150 is what you started with

tan (π/4) - Sin (3π/2)

2

Period of a trig function

2π/b, interval for one complete cycle

arcsin -1

3π/2 --> -π/2

cos^-1 (sin -π/4)

3π/4

tanx = -1

3π/4, 7π/4

Cot(cos^-1 3√34/34)

3√34/5√34

sinx = -√3/2

4π/3, 5π/3

Two fire-lookout stations are 10 miles apart, with station B directly east of station A. Both stations spot a fire. The bearing of the fire from station A is N25 degrees and the bearing of the fire from station B is N54W. How far, to the nearest tenth of a mile, is the fire from each lookout station?

5.7 mi from A and 9.2 mi from B

arccos -√3/2

5π/6

Formula for Area of a Sector

A = (1/2)*r^2*θ

Standard form

Ax + By=C, where A, B, and C are not decimals or fractions, where A and B are not both zero, and where A is not a negative

Graph of sine function

Domain: (-∞, ∞) Rage: [-1, 1] Period: 2π θ: 0, π/2, π, 3π/2, 2π y: 0, 1, 0, -1, 0

Tan/cot graph

Don't have to factor out, just set it equal to the asymptotes, which is π/2 and -π/2 for tan graphs, and 0, π for cot graphs.

Amplitude

Height of a wave

Sinθ ≤ 1, if θ>1, then it is ...

NO SOLUTION

trigonometry circle

Q1: All Functions Q2: + Sin/Csc Q3: + Tan/Cot Q4: + Cos/Sec

Formula for length of arc

S= r*θ

Right Triangle Trig (*ONLY RIGHT ANGLE*)

SOHCAHTOA

Cos graph

Same period of 2π, but y-values are 1,0,-1,0,1

Parallel Line

Same slope different y intercept

y= tan2θ

Set it equal to old asymptotes 2θ = -π/2, θ= π/4 2θ = π/2, θ= -π/4

airspeed

Speed at which the aircraft is traveling through the air; it may be less or more than the relation to the ground.

Inverse sin graph

Switches the θ to y.

y= sin(θ + π/3)

The +π/3 inside of the ( ) with the θ. It will affect horizontal change. *REMEMBER TO USE THE OPPOSITE NUMBER OF WHAT IS GIVEN!* subtract π/3 from the original θ.

angle of depression

The angle formed by a horizontal line and the line of sight to an object below the horizontal line

45-45-90 triangle

The legs are √2/2 and r = 1

y= 2sinθ

The number in front of any trig functions will change the y-values. Multiple the original y-values by the number in front of sin, and get 0, 2, 0, -2, 0.

y= sin2θ

The number in front of θ will affect the period. Now you use 2π divided by the number in front of θ which is 2 and get π. Then divided π by 4 because we need 4 units to label our graph, and get π/4 for our measurement for each points.

y= sin(-2θ + π/2)

You have to simplify the equation by moving the -2 out, which made it y= -sin (2(θ - π/4)). REMEMBER TO DIVIDE THE π/# TOO!!!

Asymptote

a line that a graph approaches but never crosses

30-60-90 triangle

across the 60 degrees is √3/2, across 30 degrees is 1/2, and r = 1

To find the midpoint (period) of tan/cot graph, ______________.

add two asymptotes and divide by two.

Law of cosines

a²=b²+c²-2bcCosA b²=a²+c²-2acCosB c²=a²+b²-2abCosC - SAS - SSS

Inverse Trig Functions

changes y to x

Cos(sin^-1 5√29/29)

cos (1.19) .37 radians

Distance formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Odd Function

f(-x) = -f(x) - csc/sin - tan/cot

Even Function

f(-x) = f(x) - cos/sec

Average rate of change

f(b)-f(a)/b-a

Convert radians to degrees

multiply by 180/π

Convert degrees to radians

multiply by π/180

perpendicular lines

negative reciprocal slopes

Sec

r/x or 1/cos

Csc

r/y or 1/sin

Law of sines

sinA/a=sinB/b=sinC/c - AAS - ASA - *ASS* Special cases

Trig Identities

sin^2 x + cos^2 x = 1 cos^2 x = 1-sin^2 sin^2 x = 1-cos^2 tan^2 x + 1 = sec^2 x tan^2 x = sec^2 x -1 1= sec^2 x - tan^2 x 1 + cot^2 x = csc^2 x cot^2 x = csc^2 x - 1 1= csc^2 x - cot^2 x

angle of elevation

the angle formed by a horizontal line and the line of sight to an object above the horizontal line

reference angle

the angle made with the terminal arm of the standard angle and the x-axis - Q1: θ - Q2: π-θ/180-θ - Q3: θ- π/θ-180 - Q4: 2π-θ/360-θ

groundspeed

the horizontal component of the speed of an aircraft relative to the earth's surface

cot 2π

undef

Cos

x/r Use *horizontal* line

Cot

x/y (1/tan)

point slope formula

y-y₁=m(x-x₁)

Sin

y/r Use *Vertical* line

Tan

y/x (sin/cos)

You can name cos graph into a sin equation too!

y= asin (bx + c) + d 1) S.A.: y=D --> Just look; Max point- min point/2 2) Amplitude: lAl --> max - S.A. ; S.A. - min; Max - min/2 3) Period: 2π/B --> Max to Max; min to min, pattern 4) Start: x= C ---> pick

Graph shifts right

y= asin (bx + c) + d c<0

graph shifts left

y= asin (bx + c) + d c>0

Vertical shift down

y= asin (bx + c) + d d<0

Vertical shift up

y= asin (bx + c) + d d>0

slope-intercept form

y=mx+b, where m is the slope and b is the y-intercept of the line.

y= cot(θ + 5π/6)

θ + 5π/6 = 0, θ = -5π/6 θ + 5π/6 = π, θ = π/6

y= tanθ

θ = -π/2, π/2 Zero will be (0,0) - go towards upward right corner

cosx = 0

π/2, 3π/2

60 degrees

π/3

arccos 1/2

π/3

45 degrees

π/4

30 degrees

π/6

arctan √3/3

π/6

cot π/3

√(3)/3

sec(sin^-1 x)

√1-x^2/1-x^2