This document discusses trigonometric relationships between related angles. It contains the following rules:
1) sin(a) = sin(180 - a) and cos(a) = cos(180 - a), meaning trig functions of complementary angles are equal.
2) sin(a) = sin(-a) and cos(a) = cos(-a), meaning trig functions of coterminal angles are equal.
3) sin(a) = cos(90 - a) and cos(a) = sin(90 - a), relating trig functions that are 90 degrees out of phase.
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Emissions from the livestock sector account for 14.5 % of all Greenhouse gas emissions - a similar amount to that produced by the entire global amount of vehicle emissions - yet we don,t hear much about the impact of the Livestock Industry - should we think of raising taxes through this route
CIRCLE
DEFINITION AND PROPERTIES OF A CIRCLE
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Foundations of Trigonometry: Navigating Angles and Ratios with Ease"abhishek2019pandey
Title: Unveiling the Basics of Trigonometry
I. Introduction
Definition of Trigonometry
Historical context and origins
Importance in mathematics and real-world applications
II. Fundamental Concepts
Definition of angles and their measurement
Introduction to right-angled triangles
Primary trigonometric ratios: sine, cosine, tangent
III. Trigonometric Functions
Definition of trigonometric functions
Graphs of sine, cosine, and tangent functions
Periodicity and amplitude
IV. Trigonometric Identities
Pythagorean identity
Reciprocal identities
Quotient identities
V. Solving Triangles
Use of trigonometric ratios to solve triangles
Application of the Law of Sines and Law of Cosines
Examples and practical problem-solving
VI. Applications of Trigonometry
Navigation and astronomy
Engineering and physics applications
Everyday scenarios demonstrating trigonometric principles
VII. Advanced Topics (Brief Overview)
Unit circle and radian measure
Trigonometric equations
Trigonometric functions of any angle
VIII. Interactive Examples and Demonstrations
PowerPoint slides demonstrating key concepts
Interactive activities for audience engagement
Real-life scenarios illustrating trigonometric principles
IX. Practical Tips and Tricks
Memory aids for trigonometric ratios
Problem-solving strategies
Common mistakes to avoid
X. Conclusion
Recap of key concepts
Emphasis on the practical relevance of trigonometry
Encouragement for further exploration and learning
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Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
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Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2. What is to be learned?
• How to use related angles to come up with
some pretty interesting rules (not!)
3. Related Angles Reminder
a0
180 – a
180 + a 360 – a
iii
iii iv
Easiest when starting
in Quadrant 1
(Acute angle)
Relations of 700 i 700
ii 180 – 70 = 1100
iii 180 + 70 = 2500
iv 360 – 70 = 2900
4. Related Angles Reminder
a0
180 – a
180 + a 360 – a
iii
iii iv
AS
T C
sin 500
= sin 1300
sin 200
= sin 1600
Rule
sin a0
= sin (180 – a)0
5. Related Angles Reminder
a0
180 – a
180 + a 360 – a
iii
iii iv
AS
T C
cos 500
= cos 1300
cos 200
= cos 1600
Rule
cos a0
= cos (180 – a)0
-
-
-
6. Remember
Angles measured anti clockwise from
horizontal
a0
Angles measured clockwise from
horizontal
-a0
-a0
= 360 – a
and there’s more
7. Related Angles With Negatives
a0
180 – a
180 + a 360 – a
iii
iii iv
AS
T C
sin -300
= sin 3300
= sin 300
Rule
sin a0
= sin (-a)0
-
-
8. Related Angles With Negatives
a0
180 – a
180 + a 360 – a
iii
iii iv
AS
T C
cos -400
=cos 3200
= cos 400
Rule
cos a0
= cos (-a)0
11. Some Exciting Trig Rules
sin a0
= sin (180 – a)0
cos a0
= cos (180 – a)0-
e.g. if sin 400
= 0.6 then sin 1400
= 0.6
if cos 400
= 0.8 then cos1400
= 0.8
sin a0
= sin (- a)0
cos a0
= cos (- a)0
-
e.g. if sin 200
= 0.3 then sin (-20)0
= 0.3
e.g. if cos 200
= 0.9 then cos(-20)0
= 0.9
-
-
12. sin a0
= cos (90 – a)0
e.g. if sin 100
= 0.2 then cos 800
= 0.2
cos a0
= sin (90 – a)0
