Download 4.1-4.5, 4.7-4.8 You can use your formulas but NOT your unit cirlce

Math Analysis Ch 4 Review
Sections on test: 4.1-4.5, 4.7-4.8
You can use your formulas but NOT your unit cirlce
4.1: Be able to convert from DMS to degrees, degrees to DMS, degrees to radians, and radians to degrees, be able to
answer any questions about arc length
4.2: be able to find the 6 trig ratios given a triangle, evaluate on a calculator
6 trig functions
sine θ = sin θ =
cosecant θ = csc θ =
cosine θ = cos θ =
secant θ = sec θ =
tangent θ = tanθ =
cotangent θ = cot θ =
SOHCAHTOA
CHOSHACAO
4.3: be able to find any of the 6 trig ratios using the following: terminal side, reference triangle, quadrantal angles,
using what you know about the quadrants to draw a triangle, and the unit circle
Terminal side: point P (a, b) is on the terminal side of angle θ. Evaluate the 6 trig functions for θ. If the function is undefined, write
“undefined”
Steps:
1. Draw a coordinate plane and plot the point
2. Draw a triangle and label your points a and b correctly
3. Use Pythagorean theorem to find r
4. Find all 6 trig ratios (if the triangle is in the 2nd, 3rd,
or 4th quadrant you can have a “-“ answer)
Reference triangle: evaluate without a calculator by using ratios in a reference triangle
Steps:
1. Convert to degrees if necessary
2. Draw the angle in the coordinate plane
3. Determine if a 45-45-90 or 30-60-90 triangle is formed
4. Evaluate the ratio, remember the x-value is always 1 (use the standard 45-4590 and 30-60-90 given the other day) – don’t forget “negative” when necessary
Quandrantal angle: use quadrantal angles to find sin, cos, and tan. If the function is undefined, write “undefined”
Steps:
1. Determine where the point is (it will end on one of the axis – 90, 180, 270, or 360)
2. Determine the coordinates of P
3. Use “definition of trig functions in any angle” to find the ratio, sin = y/r, cos = x/r, tan = y/x
Using what you know about the quadrants to draw a triangle: evaluate without a using a calculator
Steps:
1. Determine which quadrant(s) would satisfy the conditions given, then choose that quadrant
2. Draw and label your triangle in the correct quadrant (the origin will always be θ)
3. Use Pythagorean theorem to find the 3rd side of the triangle
4. Use trig ratios to find the ones that are being asked for. Don’t forget “-“ if the triangle is in quadrant 2 or 4
Use the unit circle: find the value of the unique real number θ between 0 and 2 that satisfies the 2 given conditions
Steps:
Need to use the Unit circle for these
1. Find where the 1st given trig ratio is (there will be 2 points)
2. Determine which point would satisfy the 2nd condition, the coordinating radians is your answer
4.4-4.5: be able to graph any of the 6 ratios, be able to identify their amplitude, period, frequency, domain and
range, be able to describe a transformation and to graph the transformation ( except for cosecant and secant), and
find θ given a ratio and a given interval using reference triangles
Know the following formulas: Amplitude: |a|, Period:
2
(usually will graph 2 periods, which for all parent functions is from -
), Frequency:
Graphs of the 6 parent functions
1. f(x) = sin x
2. f(x) = cos x
Ordered pairs :
(-2
(
(
(
, 1), (, (
(-
, (0, 0),
(-2
, (2
(
(
(
, 0), (-
,(
3.f(x) = tan x
Ordered pairs :
(-2
,(
,(
,(
, (-
(0, 1),
(-
(0, Ø),
(2
4. f(x) = cot x
, Ø), (,(
(-
(0, 0),
(-2
(2
,(
, 0), (-
(
5. f(x) = csc x
(
(2
, Ø), (-
(-
6. f(x) = sec x
Ordered pairs :
(-2
(
,(
, (
, 1), ((
(-
, (0, Ø),
(2
Transformations:
Are in the form: f(x) = a sin (b(x – h)+ k
Phase shift: h (horizontal shift)
Vertical translation: k (vertical shift)
Vertical Stretch: a > 1
Vertical Shrink: 0<a<1
Reflection over x- axis: -a
(-2
,(
(
(
(
(2
(0, 1),
To transform:
1) determine h and k
2) move left or right the correct # of units and then up k
3) if there is a stretch or shrink, make sure that you change your amplitude to reflect that stretch or shrink
for example if a = 2, your are doubling your amplitude but if a = ½ you are shrinking your amplitude in half (DON’T
WORRY ABOUT STRETCHES OR SHRINKS JUST YET)
4) I suggest that graph both your parent function and translation together
Solve for x in the given interval. You should be able to find these numbers without a calculator, using reference triangles in the proper
quadrants
Steps: 1) determine what quadrant you need to be in (0 < x <
2) draw the triangle and label properly
3) determine the 3rd side of the triangle (will be 1, , 2, or
depending on the reference triangle created)
4) determine the reference angle – either 30˚, 60˚, or 45˚
5) Find x in degrees
If the triangle is in quad 1: do nothing, your reference angle is x
If the triangle is in quad 2: 180 – reference angle
If the triangle is in quad 3: 180 + reference angle
If the triangle is in quad 4: 360 – reference angle
6) Convert degrees to radians
4.7: evaluate an inverse trig function w/ and w/out a calculator, composing trig and inverse trig function
sin-1 x: domain: [-1, 1] and range: [-
cos-1 x: domain [-1, 1], range: [0, ]
tan-1 x: domain: (-∞, ∞), range: [-
cot-1 x: domain: (-∞, 0)
, range: (-
Inverse functions are giving you θ not the ratio
Evaluate w/ a calculator: 1) make sure your calculator is in the correct mode
2) 2nd (sin, cos, or tan) the ratio enter
3) the answer will be the angle measure in degrees or radians
Evaluate w/out a calculator:
1) determine what quadrant you will draw your triangle in (must know the ranges for the 3 inverses to do this: arcsin:[-
arccos:
[0, , and arctan: [***
2) draw your triangle and label the sides correctly (should end up with a 30-60-90 or 45-45-90)
3) determine your reference angle (will be 30, 60, or 45, or could be negative)
4) convert to radians
***If your point is on an axis (cos-1 = 1, 0, or -1, sin-1 = 1, 0, or -1, or tan-1 = 0 or θ) then you DO NOT draw a triangle, all you need
to do is decide what axis you’re on and if the answer is : -
Composing a trig and inverse trig function:
1) Draw your triangle
2) Label your triangle - it will be a reference triangle
3) Once you find the answer to the innermost functions, rewrite your problem
4) Determine which quadrant your new triangle will be in
5) Draw that triangle and label it correctly
6) Determine θ in radians
4.8: solve word problems involving trig
Use trig ratios to solve problems involving angle of elevation or angle of depression
To do (single angle):
1. Determine your angle of elevation (angle of depression = angle of elevation)
2. Using the measurement you are given and the one you are looking for determine which trig ratio you will be using (choose
between sin, cos, and tan)
3. Write your equation
4. Solve and write out the units
To do (2 angles):
1. You will write 2 trig equations based on 2 different triangles
2. Solve for the variable they have in common
3. Then either set them equal to each other and solve or substitute in a variable and solve (I will give you an example of each)
Examples given 2 angles:
1) Set equations equal to each other:
Two people are 10 feet apart and are looking up at a tower. Person 1‘s angle of elevation of 40˚ and person 2’s angle of
elevation is 48˚. How tall is the tower?
Person 1: tan 40 =
, solve for h: (x + 10) tan 40 = h
Person 2: tan 48 = , solve for h: x tan 48 = h
Since both equations are equal to h, you can set them equal to each other:
(x + 10) tan 40 = x tan 48
x tan 40 + 10 tan 40 = x tan 48 (distribute tan 40)
10 tan 40 = x tan 48 – x tan 40 (subtract x tan 40 from both sides)
10 tan 40 = x(tan 48 – tan 40) (factor out the x)
= x (divide by tan 48 – tan 40)
x ≈ 30.90459723 (plug the above into your calculator – be very careful when dividing trig ratios and make sure you close all your
parenthesis at the appropriate place, also make sure your calculator is in degrees)
Now solve for h:
h = x tan 48 = (30.90459723)tan 48 ≈ 34.32 feet
2) Substitute:
You are standing on a rock that is 100 feet tall. You observe a boat moving towards the rock. If the angle of elevation of the
boat changes from 32˚ to 40˚ during the period of
observation, how far does the car travel?
Smaller triangle: tan 40 =
d=
(solve for d)
larger triangle: tan 32 =
d+x=
x=
(solve for d + x)
(solve for x)
x=
(substitute
for d)
x ≈ 40.86 feet (plug into the calculator to solve, be careful about parenthesis)
Review problems: answers for you to check are coming in another post
1.
2.
3.
4.
Convert to degrees: 54˚ 28’ 18”
Convert to DMS: 38.75˚
Convert to radians: 135˚
Convert to degrees: 1.83 radians
5. Convert to degrees:
6. Complete the table
s
r
θ
18 in
_____ 38˚
____ 4.5 cm
rad
18 ft
12 ft
____
7. Find the perimeter of a 25˚ of pizza with a diameter of 20 inches.
8. Find the 6 trig ratios given the triangle below:
9. If the tan = find the remaining 5 trig ratios
10. Evaluate on a calculator:
a. sin 49˚
b. cos 18˚ 24’
c. tan
d. csc
e. sec 29˚ 18’ 25”
f. cot 85˚
11. Use the terminal side of a angle θ Evaluate the 6 trig function for θ.
a. P (2, -4)
b. P (-3, 5)
12. Evaluate without using a calculator by using ratios in a reference triangle.
a. tan
b. cos 150˚
13. Use quandrantal angles to find sin θ, cos θ, and tan θ
a. -270˚
b. 4
14. Evaluate without using a calculator
a. Find sec θ and csc θ if cot θ =
b. Find tan θ and sec θ if sin θ = =
and cos θ < 0.
and cos θ > 0
15. Find the value of the unique real number θ between 0 and 2 that satisfies the 2 given conditions
a. sin θ =
and tan θ < 0
b. cos θ =
and sin θ < 0
16. Determine the amplitude, period, and frequency
a. f(x) = 2 sin (3x – 4)
b. f(x) = -3 cos (
17. graph
a. f(x) = sin x
b. f(x) = sec x
c. f(x) = csc x
d. f(x) = cos (x –
e. f(x) = tan (x +
18. Solve for x in the given interval without a calculator
a. csc x =
,
b. sec x = c. cot x =
,
19. Evaluate on a calculator (round to the nearest hundredth)
In degrees:
a. sin-1 (.23)
b. arc cos (-.54)
in radians:
c. arc tan (4.56)
d. cos-1 (.85)
20. evaluate without a calculator
a. sin-1 (
)
b. cos-1 1
c. tan-1 (- )
d. arcsin (cos (
e. cos (sin-1 (1/2))
f. tan-1 (cos )
g. arccos (tan (
21. The top row of the red seats behind home plate at Cincinnati’s Riverfront Stadium is 90 ft above the level of the playing field.
The angle of depression to the base of the left field wall is 14˚. How far is the base of the left field wall from a point on level
ground directly below the top row?
22. To determine the height of the Louisiana-Pacific (LP) Tower, the tallest building in Conroe, Texas, a surveyor stands at a
point on the ground; level with the base of the LP building. He measures the point to be 125 ft from the building’s base and
the angle of elevation to the building to be 29.8˚. Find the height of the building.
23. From the top of a 100 foot building a man observes a car moving toward him. If the angle of depression of the car changes
from 15˚ (where the car is drawn) to 33˚ during the period of observation, how far does the car travel?
24. Two guy wires attached to a tower make angles of 60˚ and 30˚ with the ground. If the cables are attached to the ground 40 m
from
each other, find the height of the tower.