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