45 Essential Trigonometry Formulas

Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles. Mastering trigonometric formulas is crucial for solving problems in various fields, including physics, engineering, and computer science. Here's a comprehensive list of 45 essential trigonometry formulas.

1. Pythagorean Identities (3)

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

2. Quotient Identities (2)

• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)

3. Reciprocal Identities (6)

• csc(θ) = 1 / sin(θ)
• sec(θ) = 1 / cos(θ)
• cot(θ) = 1 / tan(θ)
• sin(θ) = 1 / csc(θ)
• cos(θ) = 1 / sec(θ)
• tan(θ) = 1 / cot(θ)

4. Co-function Identities (6)

• sin(90° - θ) = cos(θ)
• cos(90° - θ) = sin(θ)
• tan(90° - θ) = cot(θ)
• cot(90° - θ) = tan(θ)
• sec(90° - θ) = csc(θ)
• csc(90° - θ) = sec(θ)

5. Sum and Difference Identities (6)

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

6. Double-Angle Identities (6)

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ)
• cos(2θ) = 2cos²(θ) - 1
• cos(2θ) = 1 - 2sin²(θ)
• tan(2θ) = 2tan(θ) / (1 - tan²(θ))
• cot(2θ) = (cot²(θ) - 1) / (2cot(θ))

7. Half-Angle Identities (6)

• sin(θ/2) = ±√((1 - cos(θ)) / 2)
• cos(θ/2) = ±√((1 + cos(θ)) / 2)
• tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
• tan(θ/2) = sin(θ) / (1 + cos(θ))
• tan(θ/2) = (1 - cos(θ)) / sin(θ)
• cot(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))

8. Product-to-Sum Identities (4)

• sin(A)cos(B) = 1/2[sin(A + B) + sin(A - B)]
• cos(A)sin(B) = 1/2[sin(A + B) - sin(A - B)]
• cos(A)cos(B) = 1/2[cos(A + B) + cos(A - B)]
• sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]

9. Sum-to-Product Identities (4)

• sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)
• sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)
• cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
• cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)

Further Resources

For more detailed explanations and examples, explore these resources on trigonometric identities from pakmath.com:

• Trigonometric Identities
• Pythagorean Identities
• Sum and Difference Identities
• Double and Half Angle Identities

These formulas are fundamental to trigonometry and are used extensively in various mathematical and scientific applications. Mastering them will significantly improve your problem-solving skills.

General Methods of Teaching(Qualify B.Ed) mcqs (101-125)

General Methods of Teaching(Qualify B.Ed) mcqs (101-125) consist of 25 mcqs with answers and also their gradings. Attempt these Mcqs and prepare yourself for all kinds of tests specifically for headmaster test.

General Methods of Teaching(Qualify B.Ed) mcqs (81-100)

Qualify B.Ed Mcqs 

81. Objective related to affective domain is: 

(a) Student can paint a picture 

(b) Student can draw a graph 

(c) Student values honesty 

(d) Student can write a letter 

82. Bringing together scientific ideas to form a unique idea is:  

(a) Application  

(b) Analysis 

(c) Synthesis

(d) Evaluation 

83. Which is vast in scope: 

(a) Teaching Tactic 

(b) Teaching Technique 

(c) Teaching Strategy 

(d) Teaching Method 

84. Students find/explore the informations themselves in: 

(a) Lecture method 

(b) Discovery method 

(c) Both 

(d) None

85. Teacher performs practically and explains in:

(a) Lecture method 

(b) Discovery method 

(c) Demonstration method  

(d) Problem solving method 

86. Role of student is active in: 

(a) Discovery method  

(b) Problem solving method 

(c) Inquiry method 

(d) All above 

87. Micro teaching is a:  

(a) Teaching method 

(b) Teacher training technique 

(c) Motivational technique 

(d) None of above 

88. What is the time of presentation in Micro teaching?  

(a) 1-5 min 

(b) 5- 10 min 

(c) 10 15 min 

(d) 15- 20 min 

89. What is the No. of students in micro teaching? 

(a) 1-15  

(b) 5 – 10 

(c) 10 – 15

(d) 15 – 20  

90. Microteaching started in: 

(a) 1950  

(b) 1960 

(c) 1970 

(d) 1980

91. Microteaching focuses on the 

(a) Methods 

(b) Skills 

(c) Contents 

(d) None of above

92. Which option is  more suitable in a field of teaching of science? 

(a) Lecture method  

(b) Demonstration method 

(c) Discussion method 

(d) Project method 

93. Which one  is exception? 

(a) Books 

(b) Magazines 

(c) Diagrams

(d) T.V 

94. Which is not included in print media? 

(a) Books 

(b) Magaziņes 

(c) Diagrams 

(d) T.V 

95. A person uses how many senses while observing film? 

(a) 1 

(b) 2 

(c) 3  

(d) 4 

96. Through the sense of seeing how much knowledge is gained? 

(a) 75% 

(b) 13% 

(c) 06% 

(d) 3% 

97. Through the sense of listening how much knowledge is gained? 

(a) 75% 

(b) 13% 

(c) 06% 

(d) 3% 

98. Through the sense of touch how much knowledge is gained? 

(a) 75% 

(b) 13% 

(c) 06% 

(d) 3% 

99. Through the sense of smell how much knowledge is gained? 

(a) 75% 

(b) 13% 

(c) 06% 

(d) 3% 

100. Through the sense of taste how much knowledge is gained? 

(a) 75% 

(b) 13% 

(c) 06% 

(d) 3% 

Introduction to education mcqs(131-135)

Introduction to education(131-135)

Quiz

1 / 5

1. "No system of education is better thanits teachers who serve it", is said in
   1.   Education Conference 1947
   2.   National Education Commission 1959
   3.   National Education Policy 1972
   4.   Third five year plan 1965-1970
2. It was decided that primary education upto class five will become universal by 1979 and for girls by 1982 and upto class eight for boys by 1982 and for girls by 1987
   1.   Educational Conference 1947
   2.   National Education Commission 1959
   3.   National Education Policy 1972-80
   4.   Third Five Year Plan 1965-70
3. Education Sector Reform (ESR) is an action plan of
   1.   2001-2005
   2.   2002-2006
   3.   2003-2007
   4.   2004-2008
4. Sociological roles of education are
   1.   Conservative, Critical, Creative
   2.   Creative, Critical, Evaluation
   3.   Pedagogical, Critical, Conservative
   4.   Creative, Conservative, Pedagogical
5. Elements of educational process include
   1.   Aims, Curriculum, Medium, Philosophy
   2.   Aims, Curriculum, Philosophy, Administration
   3.   Aims, Curriculum, Administration, Medium
   4.   Aim, Curriculum, Pedagogy, Evaluation

Introduction to education mcqs(121-130)

Introduction to education(121-130)

Quiz

 

1. Planned and systematic effort by an organization to provide an individual of any age a chance of education is
   
   1.   Formal Education
   2.   Informal Education
   3.   Non-formal Education
   4.   Post Formal Education
2. Formal and informal types of education are
   1.   Supplementary to each other
   2.   Opposite to each other
   3.   Compulsory for all
   4.   For children only
3. The process in which all community members learn from each other without prior planning is
   1.   Global Education
   2.   Informal Education
   3.   Universal Education
   4.   Operational Education
4. The field of study associated with exploring realities and general principles is called
   1.   Psychology
   2.   Sociology
   3.   Anthropology
   4.   Philosophy
5. Education philosophy which lays emphasis on delivery of standard curriculum through subject centred direct instruction by teacher
   1.   Perennialism
   2.   Essentialism
   3.   Progressivism
   4.   Reconstruction
6. Learning theories define learning and explain how learning takes place, hence change in learning theory chages
   1.   Pedagogy
   2.   Curriculum
   3.   Objectives
   4.   Contents
7. Field of study directly concerned with teaching learning process and improvement in student learning is
   1.   Philosophy
   2.   Sociology
   3.   Psychology
   4.   Biology
8. The main purpose of Deoband Movement was to
   1.   Preserve the islamic identity of Muslims of Sub- Continent
   2.   Motivate Muslims for independence
   3.   'Bridging gap between modern and religious education
   4.   Imparting muslims with scientific knowledge
9. Nadva Movement was established to develop and promote
   1.   Religious education only
   2.   Modern education only
   3.   Both religious and modern education
   4.   Two Nation Theory
10. The common element between Nadva Movement and Jamia Millia Islamia was emphasis on
    1.   Vocational education
    2.   Oriental Education
    3.   Elimination of differences between Muslims
    4.   Modern and religious education