trigonometry cheat sheet
TRANSCRIPT
TRIGONOMETRY
1. Find other functions of if its . (5 PTS.)
2. , find the other trigonometric functions. (5 PTS.)
3. Find the 6 trigonometric functions considering P. (6 PTS.)
4. Given that , solve for (4 PTS.)
Solve what is asked. Express your answers in simplest form. Good for 10 minutes.
End
1. Find other functions of if its
2. , find the other trig. functions.
3. Find the 6 trigonometric functions considering P
4. Given that , solve for
DAY 1
DIAGNOSTIC EXAM (TRIGONOMETRY)
SIX TRIGONOMETRIC FUNCTIONS SOHCAHTOA
PYTHAGOREAN TRIPLE
DAY 2
REVIEW QUIZ
UNIT CIRCLE
SPECIAL ANGLES
SIGN CHART OF FUNCTION ANGLES
REFERENCE ANGLE
DAY 2 REVIEW QUIZ UNIT CIRCLE 6 TRIG. FNCS OF SPECIAL ANGLES (30O, 45O, 60O) SIGN CHART OF FUNCTION ANGLES (Ang Sarap
Tumitig ni Crush) STANDARD POSITION REFERENCE ANGLE KINDS OF ANGLES (addtβl: perigon/round, conjugate
angles, coterminal angles) CONVERSION
Degree to radians Radians to degrees
REVOLUTION SYSTEM SEXAGESIMAL SYTEM
Operation on DMS (degree, minute & second notation)
RECAP
DAY 3
REVIEW QUIZ CIRCULAR SYSTEM WRAPPING FUNCTIONS QUADRANTAL ANGLES TRIGONOMETRIC FUNCTIONS OF NEGATIVE
ANGLES GRAPHS OF TRIGONOMETRIC FUNCTIONS BEARING
REVIEW QUIZ 2: SET A1. Determine a.) reference angle b.) 6 trigo. fnc. of 2. Evaluate the following: a.) 3. Determine sin A and sec A if Express your answers in simplest form. 4. Given that . Solve for the value of 5. Given: Find the other trigonometric functions.6. Convert to degrees / radians: a.) 75O b.) 270O c.) 11/6 d.) -4/57. ALTERNATIVE RESPONSE: Write TRUE or FALSE. Any form of erasure means wrong.a.) An equiangular triangle is also equilateral.b.) Secant is the reciprocal of sine .c.) The sum of all angles of any triangle is 360.d.) An angle is positive if the direction is counterclockwise.e.) 285 and 75 are coterminal angles.
REVIEW QUIZ 2: SET B1. Determine a.) reference angle b.) 6 trigo. fnc. of 2. Evaluate the following: a.) 3. Determine a.)csc A b.) sec A if Express your answers in simplest form. 4. Given that . Solve for the value of 5. Given: Find the other trigonometric functions.6. Convert to degrees / radians: a.) -210O b.) 240O c.) -7/4 d.) 7/67. ALTERNATIVE RESPONSE: Write TRUE or FALSE. Any form of erasure means wrong.a.) In a right triangle having acute angles of 30O and 60O, the length of the side opposite 30O is one-half the length of the adjacent side.b.) If the value of one function of an acute angle is known, it is possible to find the other five functions.c.) Pythagorean theorem can be applied in any kind of triangle.d.) The reciprocal function of secant is sine.e.) The shorter leg of 30O-60O-90O triangle is 1.
ANSWERS: REVIEW QUIZ 2 SET B
a.) 60b.)
2. a.) b.) c.) d.) 3. b.) 4. 5. a.) b.) c.) d.) e.) 6. a.) b.) 240O= c.)-315O d.) 210O 7. a.) FALSE b.) TRUE c.) FALSE d.) FALSE e.) TRUE
Circular system β radian(rad) is the fundamental unit- one radian is the measure of an angle, which if its vertex is
placed at the center of a circle, subtends an arc equal to the radius of the circle
From Geometry:c = 2r, if r = 1 rad
then c = 2radians, we know that c = 360O
Hence radians = 180O
s= r
From radian to degree
From degree to radian
WRAPPING FUNCTIONS
30O
12
β3
π π
=πππΆ
π π
=πππΆ
π π
=πππΆ
πΊπΌπ΄π΄π¨πΉπ
π¬πΏπ¬πΉπͺπ°πΊπ¬Find the coordinates of the circular points.
1.
2.
3.
4.
5.
π¬πΏπ¬πΉπͺπ°πΊπ¬ :πΊπ¬π» π©Find the coordinates of the circular points.
1.
2.
3.
4.
5.
π¨π΅πΊπΎπ¬πΉπΊ :πΊπ¬π» π©Find the coordinates of the circular points.
1.
2.
3.
4.
5.
QUADRANTAL ANGLES
QUADRANTAL ANGLE β terminal side of an angle in standard position coincides with one of the coordinate axes: 0O/360O, 90O, 180O, 270O
(1,0 )
π ππ0π=π¦π
=01=0
πππ 0π=π₯π
=11=1
π‘ππ0π=π¦π₯
=01=0
ππ π 0π=ππ¦
=10=β
π ππ0π=ππ₯
=11=1
π‘ππ0π=π₯π¦
=10=β
siny
r cos
x
r tan
y
x csc
r
y sec
r
x cot
x
y
0O,360O 0
01
1
11
0
01
1
0
11
1
1
0
90O 1
11
0
01
1
0
11
1
1
0
00
1
180O 0
01
1
11
00
1
1
0
11
1
1
0
270O 1
11
00
1
1
0
11
1
1
0
00
1
πΊπΌπ΄π΄π¨πΉπ
Example: Evaluate
= 450O + 540O + 630O + 720O
= 90O + 180O + 270O + 360O
= 0 + (-1) + 0 +1= 0
TRIGONOMETRIC FUNCTIONS OF NEGATIVE ANGLESTRIGONOMETRIC FUNCTIONS OF NEGATIVE ANGLES
sin (- ) = y
r
=-sin csc (- ) =
r
y= -csc
cos (- ) = x
r= cos sec (- ) =
r
x= sec
tan (- ) = y
x
= -tan cot (- ) =
x
y= -cot
Examples: Evaluate the following:
1. tan (-45O) 2. cos (-60O) 3. csc (-450O)
-
r -y
x
P (x, y)
= -1
1
GRAPHS OF TRIGONOMETRIC FUNCTIONSGRAPHS OF TRIGONOMETRIC FUNCTIONS
A. Sine function
B. Cosine function
Sine function
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180 210 240 270 300 330 360
degr ees
GRAPHS OF TRIGONOMETRIC FUNCTIONS
A. Sine function
B. Cosine function
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180 210 240 270 300 330 360
y = sin x Properties: Period = 2
Amplitude = 1 Domain = Range = [-1, 1] Nature: symmetric
with respect to the origin
GRAPHS OF TRIGONOMETRIC FUNCTIONS
A. Sine function
B. Cosine function
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180 210 240 270 300 330 360
degrees
Cosine function
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180 210 240 270 300 330 360
degrees
co
sin
e v
alu
e
GRAPHS OF TRIGONOMETRIC FUNCTIONS
A. Sine function
B. Cosine function
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180 210 240 270 300 330 360
y = cos x Properties: Period = 2
Amplitude = 1 Domain = Range = [-1, 1] Nature: symmetric
with respect to the y-axis
Tangent function
-2.75
-2
-1.25
-0.5
0.25
1
1.75
2.5
-90 -60 -30 0 30 60 90 120 150 180
degrees
tan
gen
t va
lue
y = tan x Properties: Period = Amplitude = undefined
Domain = 2 k , k is an integer
Range = Nature: symmetric with respect to the
origin Increasing function between consecutive
asymptotes
Discontinuous at 2x k , k is an
integer
PARTS OF THE GRAPH DEFINITION OF TERMS
1. Nodes β points where the curve intersects the neutral axis 2. Amplitude β absolute value of the maximum distance of the curve from the neutral axis 3. Period β duration (in degrees/radians) to complete a cycle 4. Wavelength β complete cycle
Sine function
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 30 60 90 120 150 180 210 240 270 300 330 360
degrees
nodes
period
wavelength
N.A. (neutral axis)
amplitude
PROPERTIES OF GRAPHS OF TRIGONOMETRIC FUNCTION Different Graphs Properties
1. sin
cos
y a x
y a x
Amplitude Period
a
2
2. sin
cos
y a bx
y a bx
Amplitude Period
a
2
b
(the effect of b is it stretches or compresses
the graph so that its new period is 2/b)
3.
sin
cos
y a bx c
y a bx c
Amplitude Period Phase shift End point
a
2
b
c
b(if c
b is positive, curve shifts to the right)
(if c
b is negative, curve shifts to the left)
c
b+
2
b
(starting point + period
4.
sin
cos
y a bx c d
y a bx c d
Amplitude Period
Phase shift
Translation
a
2
b
c
b
d (if d is positive, N.A. shift above the x-axis) (if d is negative, N.A. shift below the x-axis)
PRACTICE
π=βπ πππ(ππ π+ π π )βπ
Amplitude = Period = Interval = Phase shift = End point = =Translation = -1
x 180o 150o 120o 90o 60o 30o
Refer to the graph below _____ 1 . The figure describes the graph of
A. cosine function B. cosecant function C. tangent function D. cotangent function _____ 2. The number of cycles the graph has
A. 2 B. 3 C. 4 D. 5 _____ 3. The equation of the graph is
A. y = tan3x B. y = cos3x C. y = sin3x D. y = cot3x _____ 4. What is the period of the function?
A. 30o B. 60o C. 90o D. 120o _____ 5. The function has a frequency of
A. 2 B. 3 C. 4 D. 5 _____ 6. The range of the function is equal to A. -β β€ y β₯ +β B. 0 β€ y β€ +β C. 0 β€ y β€ -β D. y β₯ +β _____ 7. Which of the following does not belong to the group? A. 30o B. 90o C. 120o D. 150o _____ 8. Which of the following is NOT a zero of the function? A. 0o B. 90o C. 120o D. 180o
APPLICATION OF RIGHT TRIANGLES
A. BEARING
B. ANGLE OF ELEVATION AND DEPRESSION
BEARINGBearing - direction of one point with respect to a given point Types of bearing:
1. True/Course bearing (T)- angle measured from north clockwise 2. Simple Bearing (S)- acute angle measured from north or south
Examples:
1. 2.
35O
E
N
W
S T: S:
35O
E
N
W
S
T: S:
3. 4.
35O
E
N
W
S
T: S:
35O
E
N
W
S
T: S:
Examples: Solve the following: 1. Clarkβs house is 4 kilometer (km) N65O40βE of SM Taytay while Bruceβs house is 3 km
S24O20βE of SM Taytay. Find the distance between the two houses. 2. MV Cristina is 85 km to the East and 107 km to the south of a certain port. Find its distance
and bearing from the port. 3. Two ships left the same port at the same time, MV Katrina is going in the direction N70OE
and MV Milagros is sailing East. MV Katrina traveled at 30 kilometer per hour (kph). After 30 min, MV Milagros was observed to be directly south of MV Katrina. Find the speed MV Milagros.
4. Three ships are situated as follows: A is 250 miles due North of C, and B is 375 miles due East of C. What is the simple bearing of a.) B from A b.) A from B?
5. Determine the simple and true bearing of the figure.
O 40O
23O
C A
B
25O
ANGLE OF ELEVATION AND DEPRESSION
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
Examples: Solve the following:1. From the top of a light house, 135 meters(m) high, it is observed that the angle of depression of a ship is 21O. How far is the ship from the top of the mountain?2. Bea, standing 9m. above the ground, observes the angles of elevation and depression of the top and bottom of the Rizal monument in Luneta as 6O50β and 7O30β respectively. Find the height of the monument.3. Maru is 5 feet (ft.) tall and casts a shadow of 6 ft. on the ground. Find the angle of elevation of the sun.4. From two points each on the opposite sides of the river, the angles of elevation of the top of an 80 ft. tree are 60O and 30O. The points and the tree are in the same straight line, which is perpendicular to the river. How wide is the river?5. A mountain peak stands near a level plain on which two farm houses, C and D are in straight line from the peak. The angle of depression from the peak to C is 50O42β and the angle of depression to D is 25O30β. The peak is known to be 1,005 meters above the level plain. Find the distance from C to D.
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
1. From the top of a light house, 135 meters(m) high, it is observed that the angle of depression of a ship is 21O. How far is the ship from the top of the light house?
21O
135 m. x
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
2. Bea, standing 9m. above the ground, observes the angles of elevation and depression of the top and bottom of the Rizal monument in Luneta as 6O50β and 7O30β respectively. Find the height of the monument.
9 m.
6O50β
7O30βh
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
3. Maru is 5 feet (ft.) tall and casts a shadow of 6 ft. on the ground. Find the angle of elevation of the sun.
5 ft.
6 ft.
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
4. From two points each on the opposite sides of the river, the angles of elevation of the top of an 80 ft. tree are 60O and 30O. The points and the tree are in the same straight line, which is perpendicular to the river. How wide is the river?
x
80 ft.
x
ANGLE OF ELEVATION AND DEPRESSION PROBLEMS
5. A mountain peak stands near a level plain on which two farm houses, C and D are in straight line from the peak. The angle of depression from the peak to C is 50O42β and the angle of depression to D is 25O30β. The peak is known to be 1,005 meters above the level plain. Find the distance from C to D.
25O30β 50O42β
25O30β
50O42β
C D
P
A
1005 m.
x x
QUIZ: BEARING & ANGLE OF DEPRESSION & ELEVATION1. A plane takes off on a runway that is horizontally 915 ft. from a building, 121 ft. high. What is the minimum angle of elevation of its take off to assure of going over the building if it flies in a straight line? (3 PTS.) 2. A missile that was launched has angle of depression from the point of launch has 30O20β angle of depression and was known to be 1185 ft. away from the ground. Find the distance the missile had traveled. (3 PTS.) 3. At a considerable distance away from the base of a cliff, a surveyor found the angle of elevation to the top of a cliff to be 70O. After moving a distance of 100 m. in a horizontal line farther to the cliff, the angle became 50O. How high is the cliff? (3 PTS.) 4. An airplane traveled 60 km. with a bearing of . Due to the storm, it turned at From the starting point to its current position, the distance is 90 km. How far did it travel when it turned? (3 PTS.) 5. Determine the simple and true bearing of OA, OB, OC AND OD. (8 PTS.)
QUIZ: BEARING & ANGLE OF DEPRESSION & ELEVATION1. A plane takes off on a runway that is horizontally 915 ft. from a building, 121 ft. high. What is the minimum angle of elevation of its take off to assure of going over the building if it flies in a straight line? (3 PTS.)
121 ft.
915 ft.
QUIZ: BEARING & ANGLE OF DEPRESSION & ELEVATION2. A missile that was launched has angle of depression from the point of launch has 30O20β angle of depression and was known to be 1185 ft. away from the ground. Find the distance the missile had traveled. (3 PTS.)
1185 ft.
30O20β
x
30O20β
QUIZ: BEARING & ANGLE OF DEPRESSION & ELEVATION3. At a considerable distance away from the base of a cliff, a surveyor found the angle of elevation to the top of a cliff to be 70O. After moving a distance of 100 m. in a horizontal line farther to the cliff, the angle became 50O. How high is the cliff? (3 PTS.)
β
β‘β =β‘
β’β’toβ
h
50O 70O
100 m x
PROVING IDENTITIES
SUGGESTIONS FOR PROVING IDENTITIES
1. Learn well the formulas given above (or at least, know how to find them quickly).2. Choose the more complicated side and start transforming it so that it has the same form as the simpler side.3. Sometimes, it is more convenient to transform each side simultaneously into same equivalent form (METHOD 2).4. Try to express everything in terms of sines and cosines.5.Instead of applying suggestion 4, sometimes advantageous to convert everything into a single function only.6. Have an open mind in using algebraic processes to facilitate proving.7. You may introduce a desired factor to obtain a particular expression.8. Look for ways to use identities 6, 7 and 8 or one of its other forms. 9.Set in mind the result you want in the end.
SHORTCUT
sin cos
tan cot
sec csc
1
2 2
2
2
2
2
3.ππ π2πβππ π2ππππ 2π=1
HOMEWORK
SUM AND DIFFERENCE OF TWO ANGLES
QUIZ
LOGARITHMS