Math Unit 5 Test Flashcards
1
Q
Hypotenuse:
A
The longest side, always across from the Right Angle
2
Q
Opposite:
A
Across from the INDICATED/GIVEN angle
3
Q
Adjacent:
A
Helps form the INDICATED/GIVEN angle
4
Q
SOH CAH TOH
A
Sin 0 = Opp/Hyp
Cos 0 = Adj/Hyp
Tan 0 = Opp/adj
5
Q
There are 3 “Secondary Trig Ratios” They are…
A
the Reciprocals of the “Primary Trig Ratios”
6
Q
Cosecant
A
Csc 0 = Hyp/Opp
7
Q
Secant
A
Sec 0 = Hyp/Adj
8
Q
Cotangent
A
Cot 0 = Adj/Opp
9
Q
Angles 30°, 45° and 60° often occur in trigonometry
A
they are called SPECIAL angles
the triangles in which they are found are called SPECIAL TRIANGLES
10
Q
What does ET look like?
A
Equilateral triangle split in half
Degrees - 90 and 60 at the bottom together, and 30 at the top
Sides - √3 (adj or opp), 2 (hyp), 1 (adj or opp)
11
Q
What does IRT look like?
A
Triangle split in half
Degrees - 45, 45, 90
Sides - 1 (adj or opp), 1 (adj or opp), √2
12
Q
When the angle is 30 what is cos?
A
√3/2
13
Q
When the angle is 30 what is sin?
A
½
14
Q
When the angle is 30 what is tan?
A
1/√3
15
Q
When the angle is 45 what is cos?
A
1/√2
16
Q
When the angle is 45 what is sin?
A
1/√2
17
Q
When the angle is 45 what is tan?
A
1
18
Q
When the angle is 60 what is cos?
A
½
19
Q
When the angle is 60 what is sin?
A
√3/2
20
Q
When the angle is 60 what is tan?
A
√3/1
21
Q
From the UNIT CIRCLE diagram, we can see a rule in when RATIOS ARE POSITIVE in each of the 4 quadrants
A
This is often called the CAST RULE
22
Q
C in
A
bottom right
23
Q
A in
A
top right
24
Q
S in
A
top left
25
T in
bottom left
26
A is
I
27
S is
II
28
T is
III
29
C is
IV
30
A range
0 - 90
31
S range
90 - 180
32
T range
180 - 270
33
C range
270 - 360
34
Cosine is positive in
C
35
All ratios are positive in
A
36
Sine is positive in
S
37
Tangent is positive in
T
38
To Find the exact value of the following (use a well-labelled diagram)
Drop the perpendicular to the x-axis after drawing the angle
Subtract the max angle from each quadrant to find the value
39
For 30 just
draw IRT otherwise it confuses you lol
40
To Using the appropriate triangle, determine θ, if 0°≤ θ ≤ 360°
Figure out what special angle it is
What quadrants it is positive or negative dependingt on the trig ratio
Draw both angles in both quadrants and figure out the other angle by subtracting the og angle
41
To Find the exact value of the following. Rationalize the denominator where necessary
Use the diagram for each one
Add/Multiple together using radical rules
42
Angles
= Upper Case letters
43
Side Length
= Lower Case letters
44
Steps to Finding Side Lengths with Primary:
1. Step 1: Label the sides (Hypotenuse, Opposite and Adjacent) of your triangle relative to the given angle
2. Step 2: Determine which trig ratio to use (Sin, Cos or Tan?) by checking SOH CAH TOA
3. Step 3: SET UP the equation with the unknown side and solve for the side length
45
Steps to Finding Angles with Primary:
1. Step 1: Label the sides (Hypotenuse, Opposite and Adjacent) of your triangle relative to the Angle you want to find
2. Step 2: Determine which trig ratio to use (Sin, Cos or Tan?) by checking SOH CAH TOA
Step 3: SET UP the equation with the unknown side and solve for the angles using the inverse trig ratios
46
Elevation:
The angle is made between the HORIZONTAL and the line of sight UPWARDS to an object
47
Depression:
The angle is made between the HORIZONTAL and the line of sight DOWNWARDS to an object
48
Matching pair:
When you know the value of an ANGLE and its opposite SIDE
49
Sine law is for…
NON-RIGHT Triangles with a MATCHING PAIR
50
Sine law to find unknown SIDES…
a/Sine A = b/Sine B = c/Sine C
51
Sine law to find unknown ANGLES…
Sine A/a = Sine B/a = Sine C/c
52
Contained Angle:
When you know the value of an ANGLE and the TWO SIDES that create the angle
53
Cosine law is for…
NON-RIGHT Triangles with TWO SIDES and a CONTAINED angle
54
Cosine law to fine unknown SIDES…
c2 = a2 + b2 − 2ab cos(C)
55
You can change cosine law (sides) just always make sure that
the side you are finding has it’s matching angle at the end of the formula
56
Cosine law to find angles is when you have…
NON-RIGHT Triangles with ALL THREE SIDES
57
Cosine law to find unknown ANGLES...
cosC = a^2 + b^2 - c^2/2ab
58
You can change cosine law (angles) just swap
the matching side being subtracted for the matching angle being found
59
To SOLVE a triangle means to find:
All unknown SIDES
All unknown ANGLES
60
To find an area…
1. Use cosine to find the contained angle
2. Use SOHCAHTOA to find the height
3. Use a=bh/2 to find the area
61
To find boats/distance…
1. Find the angle in the triangle by subtracting the barrings
2. Convert the units by multiplying the km/hr by the hours to find the km
3. Use cosine law to find x
62
3-D Applications
1. Draw it
2. Find the missing angles using 180 - angle = or barrings subtract
3. Use SOH CAH TOA to find missing sides
4. Use cosine law to find the distance
63
Cscx =
1/sinx
64
1/sin =
cscx
65
Secx=
1/cosx
66
1/cosx =
secx
67
Cotx =
1/tanx
68
1/tanx =
cotx
69
Tanx =
sinx/cosx
70
sinx/cosx =
tanx
71
tan^2x =
sin^2x/cos^2x
72
sin^2x/cos^2x =
tan^2x
73
sin ^2x + cos^2x =
1
74
1 =
sin ^2x + cos^2x
75
sin^2x =
1 - cos^2x
76
1 - cos^2x =
sin^2x
77
cos^2x =
1 - sin^2x
78
Tan x =
sinx/cosx
79
Cot x = (cos&sin)
cosx/sinx
80
Cscx =
1/sinx
81
Secx =
1/cosx
82
Cotx =
1/tanx
83
sin ^2x + cos^2x =
1
84
Tan^2x + 1 =
sec^2x
85
cot^2x + 1 =
csc^2x
86
Strategies when proving trigonometric identities
1.start with the more…
Start with the more complicated side
87
Strategies when proving trigonometric identities
2.use…
Use Logical steps
88
Strategies when proving trigonometric identities
3.
Try one identity at a time and see where it takes you
89
Strategies when proving trigonometric identities:
4.
note
Note: only change a value of 1 to sin2 θ + cos2 θ as a last resort as this usually complicates things rather than simplifies them. Recall that a value of 1 can also be changed to sin/sin or cos/cos for the purpose of a common denominator
90
Strategies when proving trigonometric identities:
5.
convert waht to ehat aleays
ii) Convert tanx to sinx/cosx always
91
Strategies when proving trigonometric identities:
6. use algebra..
Use Algebra Skills
Expand
Find common denominator
Factor
92
Strategies when proving trigonometric identities:
7.always keep an…
Always keep an eye on the other side
93
On the test:
Question 1:
-Bearing of 195/155 with an angle of depression of 16degress/12
-0,90,180,270
- 50 m height
- 10 and 12-degree angles of depression
- Use cosine law
--c2 = a2 + b2 − 2ab cos(C)
--a2 = b2 + c2 - 2bccos(A)
94
On the test:
Question 2:
no bc…
No because you need the other angle of depression from the other side in order to work out the side lengths that would then allow you to finally find the distance using cosine law
95
On the test:
Question 3:
2tanx=√12
2tanx=√12
2tanx= √4√3
=2√3/2
=√3/1
=√3
96
On the test:
Question 4:
angle 1 =
angle 2 =
Angle 1 = 60
Angle 2 = 240
97
On the test:
Question 5:
Secxcscx = cotx + tanx
Secxcscx = cotx + tanx
Right side = 1/tanx + sinx/cosx
1 ➗sinx/cosx + sin/cosx
1 X cosx/sinx + sinx/cosx
cosx/sinx + sinx/cosx
cosxcosx/sincosx + sinxsinx/sinxcosx
Cos2x/sincosx + sin2x/sinxcosx
1/sinxcosx
(1/sin)(1/cos)
secxcsc
