chapter 1

PLANE and SPHERICAL TRIGONOMETRY

Chapter 1: RIGHT TRIANGLES

MATH 12: PLANE AND SPHERICAL TRIGONOMETRY

TRIGONOMETRY- deals with triangles and interrelationships

of its sides and angles.

Applications:• mechanics, surveying, geodesy• advanced mathematics, engineering and

navigation, etc.


MEASUREMENT OF ANGLES- angle formed between the intersection of two lines or two planes.


PRINCIPAL SYSTEMS1. Sexagesimal System – degrees, minutes,

seconds2. Grade System - grades3. Circular System - radians4. Mil System - mils


Some Relationships

Between Circular and Sexagesimal systems,

To convert degrees to radians, multiply the degree measure by


ILLUSTRATIONS: Convert the following from degrees to radians or vice versa.

1. 6. 2. 7. 3. 8. 4. 9. 5. 10.


ANGLES IN STANDARD FORM

Angle in standard form starts have positive axis as the terminal side.

Angles measured counterclockwise are positive. Othewise, it will be negative.

terminal side

Initial side


COTERMINAL ANGLES- angles having the same terminal side.

𝜃=50𝑜

𝜃=−310𝑜Other measures for the coterminal angles are …


ILLUSTRATIONS. Find the angle , co-terminal to the following:


LENGTH OF AN ARC

𝜃𝑠

𝑟

𝑟

By ratio and proportion,


EXAMPLES:

1. Find the length of an arc that subtends a central angle of in a circle of radius 10 m.

2. An arc of length 100 m subtends a central angle in a circle of radius 50 m. Find .


EXAMPLES:

3. Find the distance traveled along an arc on the surface of the earth that subtends a central angle of 1 minute. (

4. Find the distance traveled by the earth in one day in its path around the sun given the ff info:


EXAMPLES:

5. A truck traveling at a rate of 35 mph goes through a circular path of radius ft. Through what angle does it turn in ?

6. A town in Luzon is in latitude . Assuming the earth is a sphere of radius , find the distance of the town from the equator.


LINEAR VS ANGULAR SPEED

𝜃𝑠

𝑟

𝑟Angular speed: , where is measured in radians.

Linear Speed:

Some angular speed is measured in terms of rev/time. Note that 1 rev = .


EXAMPLES:

1. A certain wheel rotates at an average of 25 rpm (revolutions per minute). If the radius of the wheel is 30 in, find the angular speed and the linear speed.

2. A truck with 48-in tire diameter is traveling at 50 mph. ( 1 mi = 5280 ft, 1ft = 12 in)

a. Find the angular speed of the wheel in rad/ min.

b. How many revolutions per minute do the wheels make?


EXAMPLES:

3. A radial saw has a blade with a 6-in radius. Suppose that the blade spins at 1000 rpm.

a. Find its angular speed in rad/min.b. Find its linear speed.


TRIGONOMETRIC RATIOS

Consider the following right triangle:

𝜃

a

b

c

For any right triangle,

where is the hypotenuse


Then we have the following six trigonometric ratios:

sin 𝜃=𝑎𝑐

cos𝜃=𝑏𝑐

tan𝜃=𝑎𝑏

csc 𝜃=𝑐𝑎

sec𝜃=𝑐𝑏

cot 𝜃=𝑏𝑎


ILLUSTRATIONS:

1. Evaluate the six trigonometric ratios for :

2. Find the value of

5

12 𝜃

𝑥45𝑜

12


𝑥

60𝑜65𝑜

𝑥60𝑜 30𝑜

100


EXERCISES:

1. Evaluate the six trigonometric ratios for :

2. Find the value of

25

24 𝜃

1360𝑜

𝑥

2

3 𝜃


𝑥

30𝑜

5

30𝑜60𝑜

𝑥


APPLICATIONS

• Angle of elevation/ depression

Angle of elevation

Angle of depression


APPLICATIONS

• Bearing of a line N

S

EW

32𝑜

25𝑜

What are the bearing of the two lines?MATH 12: PLANE AND SPHERICAL TRIGONOMETRY

ILLUSTRATIONS:

1. From the top of the cliff which rises vertically 168.5 ft above the river bank, the angle of depression of the opposite bank is . How wide is the river?

2. A man standing 230 ft from the foot of building finds that the angle of elevation of the top of the building is . If his eye is above the ground, what is the height of the building?


ILLUSTRATIONS:

3. A surveyor found the angle of elevation of the top of a building to be . After walking towards the building, the angle of elevation measured is . How tall is the building if the device used for surveying has a height of ?


ILLUSTRATIONS:

4. From a certain point, a ship sails 45 miles due north and then proceeds westward at a speed of 20 mph. Find the bearing of the ship and its distance from the starting point four hours after it turned westward?

5. A ship sails and travels at a bearing of for two hours before changing its course to for additional 3 hours. If the ship travels at a constant speed of , how far is the ship from its original location?


EXERCISES:

1. From the top of a lighthouse above sea level, a boat is observed under an angle of elevation of . How far is the boat from the lighthouse?

2. Two strings tether a balloon on the ground, as shown. How high is the balloon?

55𝑜 70𝑜

10 m

EXERCISES:

3. From a certain point, a boat sails 30 miles due south and then proceeds eastward at a speed of 15 mph. Find the bearing of the boat and its distance from the starting point 5 hours after it turned eastward.


SPECIAL TRIGONOMETRIC FUNCTION VALUESRecall:• Angles in standard form• Coterminal Angles• Reference Angles

- acute which the terminal side makes with the always measured as positive.


Reference Angle140𝑜

40𝑜

245𝑜

65𝑜

¿180𝑜−140𝑜

¿245𝑜−180𝑜


CIRCULARFUNCTIONS

Consider the following unit circle with center at the origin and radius equal to 1.

(1,0)

(0,1)

(-1,0)

(0,-1)

Let be defined as follows

where is the terminal point’s coordinates of angle measured in standard form.


In short,

(𝑥 , 𝑦 )

𝜃

sin 𝜃=𝑦1

=𝑦 csc 𝜃=1𝑦

cos𝜃=𝑥1=𝑥 sec𝜃=

1𝑥

tan𝜃=𝑦𝑥 cot 𝜃=

𝑥𝑦

𝑥

𝑦1

𝜃


Quadrantal Angles

𝜃=0𝑜

90𝑜

180𝑜

270𝑜

360𝑜0

𝜋2

𝜋

3𝜋2

2𝜋

For quadrantal angles,.....


The following are the summary of six trigonometric ratios:

(1,0) 0 1 0 UND 1 UND

(0,1) 1 0 UND 1 UND 0

(-1,0) 0 -1 0 UND -1 UND

(0,-1) -1 0 UND -1 UND 0


Special angles

0

𝜋6

𝜋3

𝜋22𝜋

3

5𝜋6

𝜋

7𝜋6

4𝜋3 3𝜋

2

5𝜋3

11𝜋6

Angles whose denominator are 6 have a reference angle of

Angles whose denominator are 3 have a reference angle of .


Special angle

0

𝜋6

𝜋3

𝜋22𝜋

3

5𝜋6

𝜋

7𝜋6

4𝜋3 3𝜋

2

5𝜋3

11𝜋6

30𝑜

60𝑜1

√32

12


The following are the summary of six trigonometric ratios for :

2

2

- 2

- 2


Special angle

0

𝜋6

𝜋3

𝜋22𝜋

3

5𝜋6

𝜋

7𝜋6

4𝜋3 3𝜋

2

5𝜋3

11𝜋6

60𝑜

30𝑜

1√32

12



2

- 2

- 2

2


Special angle

Angles whose denominator are 4 have a reference angle of

𝜋4

3𝜋4

5𝜋4

7𝜋4


Special angle

𝜋4

3𝜋4

5𝜋4

7𝜋4

45𝑜

45𝑜

1 √22

√22




In summary,Reference QII QIII QIV Terminal

Side

Note: The sign assignments of the coordinates will vary depending on the quadrant.


ILLUSTRATIONS: Evaluate the six trigonometric ratios for each of the ff :


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