Application of Trigonometry in Real Life Situation
Mathematics is a subject that is vital for gaining a better perspective on events that occur in the natural world an in real life situation. A keen aptitude for math improves critical thinking and promotes problem-solving abilities. One specific area of mathematical and geometrical reasoning is trigonometry which studies the properties of triangles. In some of the fields such as architecture, astronomy , biology, cartography, chemistry, civil engineering, computer graphics, geophysics, crystallography and economics , some things in these fields cannot be understood without trigonometry. These proved that trigonometry really is needed in our real life situation. Trigonometry finds a perfect partner in modern architecture. It is really needed in architecture fields, without trigonometry architecture is hard to understand. The beautifully curved surfaces in steel, stone and glass would be impossible if not for the immense potential of this science. So how does this work actually. In fact the flat panels and straight planes in the building are but at an angle to one another and the illusion is that of a curved surface. Although it is unlikely that one will ever need to directly apply a trigonometric function in solving a practical issue, the fundamental background of the science finds usage in an area which is passion for many is music. As we may be aware sound travels in waves and this pattern though not as regular as a sine or cosine function, is still useful in developing computer music. A computer cannot obviously listen to and comprehend music as we do, so computers represent it mathematically by its constituent sound waves. And this means that sound engineers and technologists who research advances in computer music and even hi-tech music composers have to relate to the basic laws of trigonometry.

Besides that, trigonometry is an arty science that can be used to measure the heights of mountains. We...

...Trigonometric Identities
I. Pythagorean Identities
A. [pic]
B. [pic]
C. [pic]
II. Sum and Difference of Angles Identities
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B. [pic]
C. [pic]
D. [pic]
E. [pic]
F. [pic]
III. Double Angle Identities
A. [pic]
B. [pic]
=[pic]
=[pic]
C. [pic]
IV. Half Angle Identities
A. [pic]
B. [pic]
C. [pic]
6-1 Inverse Trig Functions p. 468: 1-31 odd
I. Inverse Trig Functions
A. [pic]
B. [pic]
C. [pic]
Find the exact value of each expression
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic]
Use a calculator to find each value.
7. [pic] 8. [pic] 9. [pic]
Find the exact value of each expression.
10. [pic] 11. [pic] 12. [pic]
6-2 Inverse Trig Functions Continued p. 474:1-41 odd
I. Inverse Trig Functions
A. [pic]
B. [pic]
C. [pic]
Find the exact value of each expression.
1. [pic] 2. [pic] 3. [pic]
4. [pic] 5. [pic] 6. [pic] 7. [pic]
Find the exact value of each.
8. [pic] 9. [pic] 10. [pic]
Use a calculator to find each value.
11. [pic] 12. [pic]
Trigonometric Identities Trig Identities Worksheet: 1-6 all, 9, 13, 15, 19
I. Reciprocal...

...Module 5 Circular Functions and Trigonometry
What this module is about
This module is about trigonometric equations and proving fundamental identities. The lessons in this module were presented in a very simple way so it will be easy for you to understand solve problems without difficulty. Your knowledge in previous lessons would be of help in the process
What you are expected to learn
This module is designed for you to: 1. state the fundamental identities 2. prove trigonometric identities 3. state and illustrate the sum and cosine formulas of cosine and sine 4. determine the sine and cosine of an angle using the sum and difference formulas. 5. solve simple trigonometric equations
How much do you know
A. Answer the following: 1. Which of the following does not equal to 1 for all A in each domain? a. sin2 A + cos2 A
c.
b. sec2 A - cos2 A d. tan A cot A
sin A sec A
2. Simplify cos2 A sec A csc A
3. If sin ∝ =
12 4 and cos β = , where ∝ and β are both in the first 13 5 quadrant, find the values of cos (∝ + β ).
4. Sec A is equal to a. cos A b. sin A c.
1 cos A
d.
1 . sin A
5. Express
1 − csc B in terms of cos B and Sin A. cot B 1 − sin B a. cos B – sinB b. c. sin B – cos B cos B cos φ . sin φ cot φ
b. tanφ c. –csc φ d. -1
d.
sin B − 1 cos B
6. Simplify a. 1
7. Multiply and simplify ( 1 – cos2 t ) ( 1 + tan2 t ). 8. Express tan B ( sin B + cot B + cos B ) in terms of...

...CERAE
CHAPTER 6 CIRCULAR FUNCTIONS AND TRIGONOMETRY
CONTENTS
-Angles and Their Measures
-Degrees and Radians
-Angles in Standard Position and Coterminal Angles
-Angles in a Quadrant
-The Unit Circle
-Coordinates of Points on the Unit Circle
-The Sine and Cosine Function
-Values of Sine and Cosine Functions
-Graphs of Sine and Cosine Functions
-The Tangent Function
-Graph of Tangent Function
-Trigonometric Identities
-Sum and Difference of Formulas for Sine and Cosine
-TrigonometricFunctions of an angle
-Values of the Function of an Angle
-Simple Trigonometric Equations
-Right Triangle Trigonometry
-Angle of Elevation/Depression
- Solving Right Triangles
-The Law of Sines
-The Law of Cosines
CERAE
CHAPTER 6 CIRCULAR FUNCTIONS AND TRIGONOMETRY
EXPERIENCE
What I Have Learned?
I learned many things in trigonometry especially in chapter 6 , I learned many lessons specifically in the lessons 6.1-6.4 (Angles and Their Measures, Degrees and Radians, Angles in Standard Position and Coterminal Angles, Angles in a Quadrant, The Unit Circle, Coordinates of Points on the Unit Circle, The Sine and Cosine Function , Values of Sine and Cosine Functions, Graphs of Sine and Cosine Functions, and The Tangent Function).
How Did I...

...s.3 Proving Trigonometric Identities
When listening to Mr. Burger on how to prove trig identities he stated that you mightwant to work with both sides and come to common end statement. I think of proving trigidentities the same way you did proofs in geometry. You typically want to work with oneside, massage it, and hopefully you will create the expression on the other side of theequal sign. These types of problems should be viewed as ‘given a problem and itsanswer, how do you get to the answer?’. You want to keep in mind all of the trigidentities you have been exposed to thus far to assist you in proving trig identities.
Example
Prove:1tantan1cot
θ θ θ
+=+
By examining both sides of the equal sides, it appears that you want to begin with the leftside in order to create the right side.sin11tancoscos1cot1sin
θ θ θ θ θ θ
++=++
=cossincoscossincossinsin
θ θ θ θ θ θ θ θ
++
=cossincossincossin
θ θ θ θ θ θ
++
=cossinsincoscossin
θ θ θ θ θ θ
+ +
÷
=cossinsincossincos
θ θ θ θ θ θ
++
·
=sintancos
θ θ θ
=
The reason why it was best to convert in terms of sine and cosine is because the resultanttan
θ
is a trig function that can be expressed that way. There will be times when you willhave to begin with the right side of the equal sign and work your way to create the leftside of the equal sign.
Try the following:
Prove.1.
1cossintancos
x x x
− =
2.
cottanseccsc
α α α α ...

...3.3 Derivatives of TrigonometricFunctions
Math 1271, TA: Amy DeCelles
1. Overview
You need to memorize the derivatives of all the trigonometricfunctions. If you don’t get them
straight before we learn integration, it will be much harder to remember them correctly.
(sin x)
=
cos x
(cos x)
=
− sin x
(tan x)
=
sec2 x
(sec x)
=
sec x tan x
(csc x)
=
− csc x cot x
(cot x)
=
− csc2 x
A couple of useful limits also appear in this section:
lim
θ→0
lim
θ→0
sin θ
=1
θ
cos θ − 1
=0
θ
2. Examples
1.) Find the derivative of
g(x) = 4 sec t + tan t
We use the derivatives of sec and tan:
g (x) = 4 sec t tan t + sec2 t
2.) Find the derivative of
1 + sin x
x + cos x
Since y is the quotient of two functions we ﬁrst use the quotient rule:
y=
y =
(1 + sin x) (x + cos x) − (1 + sin x)(x + cos x)
(x + cos x)2
Evaluating the derivatives we get:
y =
(cos x)(x + cos x) − (1 + sin x)(1 − sin x)
(x + cos x)2
Simplifying the numerator:
y
=
(x cos x + cos2 x) − (1 − sin2 x)
(x + cos x)2
=
x cos x + cos2 x − 1 + sin2 x
(x + cos x)2
We now use the trig identity sin2 + cos2 = 1:
y =
x cos x
x cos x − 1 + 1
=
2
(x + cos x)
(x + cos x)2
2.) Find the derivative of
y = x sin x cos x
Since y is a product of functions we’ll use the product rule. We have to use it twice,
actually,...

...Submitted By:
Ma. Karla Rachelle Ulibas
Student
Submitted To:
Mr. Ray-ann Buenafe
Instructor
HISTORY OF TRIGONOMETRICFUNCTIONSTrigonometricfunctions seem to have had their origins with the Greek’s investigation of the indirect measurement of distances and angles in the “celestial sphere”. (The ancient Egyptians had used some elementary geometry to build the pyramids and remeasure lands flooded by the Nile, but neither they nor the ancient Babylonians had developed the concept of angle measure). The word trigonometry, based on the Greek words for “triangle measure”, was first used as the title for a text by the German mathematician Pitiscus in A.D. 1600.
While the early study of trigonometry can be traced to antiquity, the trigonometricfunctions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD).
The functions sine and cosine can be traced to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[23]
All six trigonometricfunctions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. Al-Khwārizmī...