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 (addt’l: 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 – 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

√3

𝝅𝟔

=𝟑𝟎𝑶

𝝅𝟒

=𝟒𝟓𝑶

𝝅𝟔

=𝟔𝟎𝑶

𝑺𝑼𝑴𝑴𝑨𝑹𝒀

𝑬𝑿𝑬𝑹𝑪𝑰𝑺𝑬Find the coordinates of the circular points.

1.

2.

3.

4.

5.

𝑬𝑿𝑬𝑹𝑪𝑰𝑺𝑬 :𝑺𝑬𝑻 𝑩Find the coordinates of the circular points.

1.

2.

3.

4.

5.

𝑨𝑵𝑺𝑾𝑬𝑹𝑺 :𝑺𝑬𝑻 𝑩Find the coordinates of the circular points.

1.

2.

3.

4.

5.

QUADRANTAL ANGLES

QUADRANTAL ANGLE – terminal side of an angle in standard position coincides with one of the coordinate axes: 0O/360O, 90O, 180O, 270O

(1,0 )

𝑠𝑖𝑛0𝑂=𝑦𝑟

=01=0

𝑐𝑜𝑠0𝑂=𝑥𝑟

=11=1

𝑡𝑎𝑛0𝑂=𝑦𝑥

=01=0

𝑐𝑠𝑐 0𝑂=𝑟𝑦

=10=∞

𝑠𝑒𝑐0𝑂=𝑟𝑥

=11=1

𝑡𝑎𝑛0𝑂=𝑥𝑦

=10=∞

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

𝑺𝑼𝑴𝑴𝑨𝑹𝒀

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 – points where the curve intersects the neutral axis 2. Amplitude – absolute value of the maximum distance of the curve from the neutral axis 3. Period – duration (in degrees/radians) to complete a cycle 4. Wavelength – 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

𝒚=−𝟑 𝒔𝒊𝒏(𝟏𝟐 𝒙+ 𝝅𝟖 )−𝟏

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. -∞ ≤ y ≥ +∞ B. 0 ≤ y ≤ +∞ C. 0 ≤ y ≤ -∞ D. y ≥ +∞ _____ 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. Clark’s house is 4 kilometer (km) N65O40’E of SM Taytay while Bruce’s house is 3 km

S24O20’E 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 6O50’ and 7O30’ 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 50O42’ and the angle of depression to D is 25O30’. 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 6O50’ and 7O30’ respectively. Find the height of the monument.

9 m.

6O50’

7O30’h

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 50O42’ and the angle of depression to D is 25O30’. The peak is known to be 1,005 meters above the level plain. Find the distance from C to D.

25O30’ 50O42’

25O30’

50O42’

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 30O20’ 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 30O20’ 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.

30O20’

x

30O20’

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.)

①

②①=②

③③to①

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.𝑐𝑠𝑐2𝜃−𝑐𝑠𝑐2𝜃𝑐𝑜𝑠2𝜃=1

HOMEWORK

SUM AND DIFFERENCE OF TWO ANGLES

QUIZ

LOGARITHMS