Math 1303 Plane Trigonometry (Measurement of Triangles)

Math 1303 Plane Trigonometry (Measurement of Triangles)

Advise: Learn(memorize) definitions and identities. Ask for help immediately. No question is too small, no question is ever

dumb – not asking is. Attempt the homework – the first step of success is an attempt to understand.

Day 1: May 28, 2002

1. Your thoughts ( definition ) of trigonometry:

Sect. 1. Review:

Geometry:  

point ( P ), ray ( AB ), line ( AB )

angle: initial side, terminal side,

We write an angle as: ______

measure of an angle: look at two ways of writing the measure of an angle

radian: to be discussed later for now 

degree: central angle that is 1/360 of a circle,

positive if measured counter clockwise, negative if measured clockwise

Types of angles ( with respect to its measure):

right:______straight: ______acute ______obtuse ______

pair of angles are

complementary if ______

supplementary if ______

Properties of triangles:

sum of the lengths of any two sides must exceed the length of the third

angles must add up to ______

triangles are similar if :

triangles are congruent if :

What is the sum of measure of the angles of a four-sided polygon ? ______

Of an n-sided polygon ? ______

Special Triangles:

right triangle: one angle that has measure ______acute triangle : all angles have a measure ____

obtuse triangle: one angle with measure ______

Pythagorean Theorem:

Given a right triangle ABC with corresponding sides abc

A Construction: Proof of Pythagorean Thm.

ex. Given any right triangle ABC ( We will label C as the right angle ). Find the missing sides

ex. A cable is to connect two points; a point at the top of a cliff to one at the base of the cliff. What length of cable

needs to be made if a point is 200 feet from the foot of a cliff and the cliff is 400 feet high.

To begin learning about the definitions of triangles and their properties we study some special angles( triangles).

1) axis angles: 90o, 180o, 270o, 360o, 0 o, ….

2) 30o – 60o - 90o right triangle and 3) the 45o – 45o – 90o right triangle

30-60 – 900 right triangle

How high above the ground can a ladder reach if it is 20 feet long and it leans against a wall making a 300 with the

ground. for safety purposes. ?

A person is running along a road that is directly below a cable car lift. The cable car makes a 60o with the

horizontal and is descending at a rate of 24 feet / sec. If the man is 480 feet away from the lift off point, then how

fast does he have to move to reach the point at the same time as the cable car. How long will it take him to get

there ?

Solving a triangle means to find all of the sides and all of the angles.

Find the missing side.

450 – 450 - 900

Find the length of the diagonal of the following “square pond” that is 40 feet wide and 40 feet long

A person stands at the edge of a river. He looks across to a point on the other side of the river. Find the width of the

river from the following picture.

Find the missing side of each triangle.

Tell me what property each of the following triangles have in common

(c.s)

(s.t.)

end of day 1

Day 2: May 29, 2002

Algebra Review:

Sect. 1.2 The rectangular coordinate system.

Directed line, directed line segment

Point P(x,y ): x coordinate or the abscissa, y coordinate or the ordinate. Radius vector: the distance from origin to

Point P.

Notice the signs on the x and y coordinates in each quadrant.

QI:QII: QIII:QIV:

------

x/r

------

y/r

------

x/y

------

Distance Formula:

To help us sketch curves we may use x-intercept, y-intercept.

ex. 2x – y = 4

ex. Sketch the graph of y = x2 + x

More Algebra Review:

Factoring:

1) GCF: x2 – 4x = ______2) trinomial: x2 + 4x – 5 = ______

3) Difference of Squares: 4x2 - 9 = ______

4) Sum/Difference of cubes: x3 + 8y3 = ______

Solving Equations, fractions: Factor, find common denominators

Quadratic Equations, Quadratic Formula:

Brief Introduction to Relations and Functions: vertical line test

All functions are relations but not all relations are functions.

Trigonometry:

There is a difference between a geometric angle and a trigonometric angle.

Note: 200 o and - 160o are coterminal but not equal neither are 10o and 370o.

Terminology:

Def. ( standard position ). An angle  is said to be in standard position provided its vertex is at its origin and its initial side

is on the positive (side) x-axis.

Note: An angle in standard position is said to be in quadrant I if its terminal side is in quadrant I. Same for other quadrant

angles.

True or False. Every quadrant I angle is acute. ______Every quadrant I angle is positive. ______

Note: If an angle is in standard position and its terminal side lies on one of the axes, we call it an axis angle.

Def (coterminal) Let  and  be any two given angles in standard position. We say that  and  are coterminal if they have

the same terminal side.

Find two angles that are coterminal with 100o.

We can define a function and a relation in terms of ordered pairs.

Def. A function is a relation in which no two ordered pairs of points have the same 1st coordinate but different 2nd

coordinates.

Six trigonometric functions – ratios of sides of a right triangle.

Def. Let P(x,y) be a point on a plane. Let r represent the radius vector of P. We define the six trigonometric functions as

follows;

sin  (sine of  ) = y/r, csc  ( cosecant of  ) = r/y,

cos  ( cosine of  ) = x/r, sec  (secant of  ) = r/x,

tan  ( tangent of  ) = y/x,cot  ( cotangent of  ) = x/y

All we need to find the six trig. functions are P(x,y) and r.

ex. Find the six trig. functions if P ( 12, 5 ).

ex. Find the six trig. functions of

(- 90o ) 60o 45o

sin 

------

cos 

------

tan 

------

end of day 2.

Day 3. May 30, 2002

Notice that r is always positive, x is positive in quadrant I and IV, and y is always positive in quadrant I and II.

So, what sign does each trig. function have in each quadrant

QIQIIQIIIQIV

sin  = y/r

------

cos  = x/ r

------

tan  = y/x

------

ex. Given cos  = -3/5 and  is in quadrant III find the remaining trig functions of .

Note: Does it matter what point we use on the terminal side of  ?

ex. Let P( 3, 4) and Q ( 12, 16) be points on the terminal side of some angle . Look at the trig functions of .

Consequences of the definitions:

1. (reciprocal functions)

2. Any trig. function of an angle is equal to the same trig. function of all angles coterminal with it.

3. The signs of the trig functions are now determined in each quadrant.

Also,

Def. We pair functions as follows and call each pair of functions cofunction of each other.

4. Any trigonometric function of an acute angle is equal to the cofunction of its complementary angle.

Alternate definitions of trig. functions.

Give you the following right triangle and I ask you to find sin  . Answer? ______

What about tan  ?

Def. Define the trig functions in terms of the words opposite, adjacent, and hypotenuse with respect to some angle .

sin  = ______cos  = ______tan  = ______

csc  = ______sec  = ______cot  = ______

Ex. Find the trig. functions of each of the following angles.

Identities – conditional equations – contradictions

x4 = x x x x , 4x = x + x + x + x ,…  identities

4 + x = 2  conditional equation

x + 4 = x - 4, sin  cos  = tan   contradiction

Fundamental Trig. Identities

1. We have already mentioned reciprocal functions.

1) csc  = ______2) sec  = ______3) cot  = ______

2. Also have ratio identities and Pythagorean identities.

4) tan  = ______5) cot  = ______

This eight identities form what we call the fundamental trig. identities.

examples:

Find the reciprocal of - 4/3  ______

If sin  = 3/7 , then what is csc  ? ______What is cos  ? ______Why ? ______

If sin  = -4/5 and cos  = 3/5, then what is tan  ? ______What quadrant is  in ? ______

Given cos  = - 3/7, What is cos2 ? ______

Algebraic techniques with Trigonometric functions

a) sin  cot  = ______b) sin2 / ( 1 – cos  ) = ______

c) 1/sin  + 1/cos = ______d) ( 2 sin  - 1 )2 = ______

To Prove Identities:

1) Work each side independently from each other – it is not an equation

2) use algebraic rules, other identities to convert from one form to another – common denominator , reduce fractions, factor,…

3) If need to , change in terms of sines and cosines .

ex. Prove or Disprove

1) cos  csc  tan  = 1

csc  tan 

2 ) ------= 1

sec 

sin 2

3) sec  - cos  = ------

cos 

sin2 - cos2 - 2 sin 

4) ------= sin  - 1

3sin  + 1

Def. We pair functions as cofunctions:

sin , cos  ; sec  , csc  ; tan , cot 

Ex.

Find sin . ______What about cos  ? ______. Is there any relationship between  and  ?

Property(of cofunctions).

A trig. function of an angle is always equal to ______

ex. Special angles(acute): 0o, 30o, 45o, 60o, 90o

 / 0o / 30o / 45o / 60o / 90o
sin 
cos 

Find each of the following values.

ex. 1)( sin 30o + cos 30o ) 2 = ______2) sin 3 30o = ______

ex. sin 20o = p, without a calculator find sin 70o = ______cos 20o = ______

tan 90o = ______sec 452 = ______

A tree casts a 30-foot shadow. The boundary line between the sunlight and the shade in midair makes an angle of 60o with the ground. How tall is the tree to the nearest foot ?

end of day 3

Day 4: May 31 ( probably day 5 as well – to the end of page 17)

Scientific notation:

decimal expression of a number in the form -- n. yyyy  10m, where 1 n  9

Conversion:

Accuracy:

We say 342 has a three figure accuracy , we say 3004 has a four figure accuracy

What about these numbers

3000  ______.

In scientific notation: 3 x 103  ______3.0 x 103  ______3.00 x 10 32 ______

With degrees we say

Significant digits / accuracy of sides example / accuracy of angles example
Two significant digits
Three significant digits
Four significant digits

Solving Right Triangles - finding all of the sides and all of the angles.

We can solve a right triangle if we are given

1) Two of the sides or 2) one of the sides and one acute angle.

Find angle A. in triangle ABC with altitude CD of length h and angle B of measure 45o.

In the following figure

If C = 26o, r = 20, find x

Use the given two triangles drawn at 90o to each other. If angle ABD = 27o, C = 62o, and BC = 42, find x and then find h.

Applications.

Angle of elevation and depression -

Bearing of a line L - acute angle formed by the north-south line and the line L.

Examples –

1. #2 page 91 an equilateral triangle has an altitude of 4.3 inches. Find the length of the sides.

Can you construct a formula for the side of a general equilateral triangles with altitude h ?

2. (6/92) A road up a hill makes an angle of 5.1o with the horizontal. If the road from the bottom of the hill to the top of the

hill is 2.5 miles long, how high is the hill.

3. (11/92) a person standing 150 cm from a mirror notices that the angle of depression from his eyes to the bottom of the mirror

is 12o, while his angle of elevation to the top of the mirror is 11o. Find the vertical dimension of the mirror.

4. (#16/ 93) A man wandering in the desert walks 2.3 miles in the direction S 31o W. He then turns 90o and walks 3.5 miles in

the direction N 59o W. At that time , how far is he from his starting point, and what is his bearing from his starting point.

ex.4 –

Find the distance between Stacey and Amy using the given topographic map on page 86.

Use the fact that the distance between Amy and Stacey on the map is 3/8// .

ex. Find the height of the flagpole if you have the capability to measure angles. See example 7 page 89.

ex. Look at example 8 page 90.

We have talked about angle of elevation, depression, bearing of a point or a line, heading. Now, let’s talk about vectors.

Vectors: magnitude and direction: force, velocity, acceleration Scalars: have only magnitude

Vectors are equal if

Addition and Subtraction of vectors –

Notation:

heading: begin from the north line and move an angle  that is measured clockwise

air speed: speed before wind acts on object ground speed: speed after wind acts on object

Components of a vector : a horizontal and a vertical vector can be added to end to obtain the original vector

Examples:

1. (14/ 108 ) A boat is crossing ariver that runs due north. The heading of the boat is due east, and it is moving through the

water at 12.0 mph. If the current of the river is a constant 3.25 mph, find the true course of the boat.

2. (18/108) A plane headed due east is moving through the air at a constant 180 mph. Its true course, however, is 65.0o. If

the wind currents are moving due north at a constant rate, find the speed of these currents.

3. ( 22/108) A ship is 2.8o off course. If the ship is traveling at 14mph, how far off course will it be after 2 hours ?

4. (34/109) A bullet is fired into the air wit an initial velocity of 1800 ft/sec at an angle of 60o from the horizontal. Find the

horizontal and vertical vectors of the velocity.

End of day 5 -

Name ______Math 1303.010 - May 29, 2002 – Quiz

1. Draw a right triangle. Label the three vertex angles (vertices ) and the three sides.

Use your labeling to state the Pythagorean Thm.

2. Through how many degrees does the minute hand move in 15 minutes ? ______

In 60 minutes ? ______

3. A trig. angle is said to be positive if it is measured in what direction ? ______

4. What is the measure of an obtuse angle ? ______

5. Complementary angles are ACUTE angles whose measures add up to ? ______

6. If two triangles have corresponding pairs of angles equal to each other, we say the triangles are ______

7. Find the missing side.

8. Complete the following triangle

9. Complete the following triangle

10. A ladder is to reach a window on the third floor. If the window is 24 feet above the ground and for safety reasons the

angle created by the ladder and the ground can not exceed 30o, then what is the shortest ladder that can be used so that

the top of the ladder reaches the window ?

Name ______Math 1303.010 – January 25, 2002 ------( Long Quiz 20 points )

1. Use the letter x,y, and r to define

a) sin  = ______b) tan  = ______c) sec  = ______

2. Find two angles that are coterminal to  = 40 o and their absolute value is less than 400o .

______

3. A forty-foot ladder leans against a wall. It touches the a point on the wall that is 20 feet above ground level. What angle

does the ladder make with the wall ?

4. Find the missing side of the following triangle.

5. Identify as a function or just a relation.

6. In what quadrant are both x/y and y/ r positive ? ______

Name ______Math 1303 - Qz # 1, January 18, 2001

HW #2 p.

Draw an angle . Label the initial and terminal sides.

Which of these angles represent a positive measure angle ? ______

Find an angle that is supplementary to 80o ______

4. If  and  are complementary, then find the sum of  and . ______

5. Draw a rt. triangle. Label the vertices(ABC) and the sides(abc). State Pythagorean Thm.

6. Find the missing sides of each of the following right triangles.

a. b.

8. Find the sides of each of the following triangles.

10. Find the measure of angle .

Name ______Math 1303.010 - January 23, 2002 –SHORT Quiz ( 10 points )

1. In what quadrant or quadrants is

a) x/y > 0  ______b) x/r < 0  ______

2. Given the points P(2, 5) find the radius vector (length) r from the origin to P.  ______

3. Find the distance between

a) A(0. – 4 ) and B ( -3, - 4 )  ______

b) P( 3, 3) and Q(-1, 6 )  ______

4. Give me a rough sketch of each of the following curves.

a) y = 2x + 4 by finding the x and y intercepts.

b) y = x2 + 2 by finding the y-intercept ,

the vertex ( -b/2a, y ) and the direction of its opening

5. What quadrant is the angle - 300o in ? - 300o is a quadrant ______angle .

6. Draw the angle 400o in standard position.

Name ______Math 1303 – January 28, 2002 --- Short Quiz ( 10 points )

1. Find an angle that is complementary to 60o ______

2. Give me an angle  so that |  | < 360o and  is coterminal with 240o . ______

3. What is the reciprocal function of sin  ? ______

Write two reciprocal functions as an equation. sin  = ______

4. Complete the following right triangle.

5. A wire is attached from the top of a 60 foot tower to the ground. The wire is grounded 60 feet from the foot of the tower.

What is the length of the wire to the nearest foot ?

______

6. Define cos  in terms of x, y , and r.

7. If P( 12, - 5) is a point on the terminal side of , then find sin  ______

8. If the angle -240o is in standard form , then what quadrant is - 240o in ? ______

9. Find a point on the terminal side of 90o. ______

10. The sun rises at 6:30 AM and sets at 6:30 PM. Assume the path to be part of a circle, through how many degrees does

the sun move in 1 hour with respect to the earth.

______

Name ______Math 1303 - January 30, 2002 --- Short Quiz 10 points

1. State the definition of cos  in terms of x, y , and r. cos  = ______

2. Use the words opp., adj., and hypotenuse to define

a) sin  = ______cot  = ______

3. State the three reciprocal identities in the proper form .

a) csc  = ______b) sec  = ______c) ______= ______

4. State the two ratio identities in the proper form.

a) tan  = ______b) cot  = ______

5. State the three Pythagorean identities.

a) sin 2 + ______= ______

b) 1 + ______= ______

c) ______+ ______= csc2

HW: Day 4 – Jan. 23, 2002

January 23: page 35: 9, 15, 27, 30, 31, 35, 39, 41, 43, 45, 47, 51, 61, 63, 66, 71

January 25: page 43: 1, 6, 7, 9, 13, 15, 18, 20, 23, 26, 27, 35, 41, 45,

LQ-long quiz

Name ______Math 1303 – Quiz # 2 - January 18, 2002

1. Find an angle that is complementary to 40o . ______

2. If  and  are supplementary what is  +  = ? ______

3. Find the missing side of each of the given triangle

a)b)

4. A ladder leans against a building. The ladder is 30 feet long and it makes an angle of 60o with the ground. How far is the

bottom of the ladder from the building (bottom) .

______

5. A person is running at 5 miles per hour. What distance will he travel in 24 minutes ? ______

6. True or False.

______a) Congruent triangles can be of different sizes

______b) Similar triangles can be the exact same size.

7. Which of these angles has a positive measure ?

8. Plot the points A( 3, - 2) , B( 0, - 4 ) and label the quadrants.

9. Find the quadrant or quadrants in which

a) x/r is positive. ______b) y/r is negative ______c) x/y is positive . ______

Name ______Math 1303 – February 1, 2002 - Long Quiz

1. Which of these three relations also represents a function ?

2. Draw the following circle  x2 + y2 = 4

3. Write the definition of sec  in terms of opposite, adjacent, and hypotenuse.

4. Write the reciprocal identity for csc  . ______

5. Write the ratio identity for tan  . ______

6. What does coss2 + sin 2 = ? ______

7. If  and  are complementary and sin  = 3/5, then find

a) csc  . ______b) cos  = ______

c) cos  . ______

8. A bird flies off the ground onto a 40 foot tree that is 60 feet away. If it flew in a straight path,

what angle does its flight line make with the ground ?

9. Reduce each of the following .

a) (sin 2 + sin  ) / ( 1 + sin  ) =

b) ( sin 2 - cos 2 ) / ( sin  + cos  )

10. Prove or Disprove.

a) csc  sec  tan  = sec 2

b) cos  - sec  = - sin2 / cos 

Math 1303 – page 80 58a, 68 page 91: 17, 21, 23, 26, 29, 33, 34, 38, 41, 98

page 107 1, 5, 13, 16, 17, 19, 20, 21, 33, 35

Name ______Math 1303 - Quiz # _____ ( 5 ? ) February 1, 2001

1. Write the definition of sec  ______

2. What is the cofunction of cos  ______

3. What is the reciprocal function of sec  ______

4. Draw  = -20o in standard position

5. Complete the sides of the following two triangles

a. b.

6. Write the three Pythagorean identities.

______

7. Write the two ratio identities.

______

8. From the top of a forty foot building a student spots a tree. The angle of depression from the building to the top of the tree

is 70o. The angle of depression from the top of the building to the bottom of the tree is 74o. How tall is the tree ?

Assume the tree is located 40 feet from the building.

5. Prove

sec  - cos  = sin2 sec

Math 1303 – TI review – some possible type of questions

1. An angle A is said to be in ______provided its vertex is at the origin and its initial

side is on the x-axis.

2. A positive measure angle is an angle that :

3. Two angles A and B are said to be ______provided ______

4. Two acute angles whose sum is 90o are said to be ______

5. Supplementary angles are ______