GK SCIENTIST

General Knowledge One Stop Source
Menu
  • Home
  • Social Science
    • History
    • Political Science
    • Geography
  • Science
    • Physics
    • Chemistry
    • Biology
  • Chemistry Notes
  • Mathematics
  • Computer
  • Tutorial MySQL
  • Tutorial Python
  • Java Tutorial
  • English Grammar
  • English Essay
  • Indian Anthropology
  • Social Anthropology
  • Philosophy
  • Solved Paper
  • UPSC
  • Current Content
    • Current Affairs
    • RSTV News
    • Yojana and Kurukshetra Gist
  • Donate
  • Contact Us

If you are interested in advertising to our audience, submit the advertising enquiry form.

Advertising Enquiry
Mathematics

Trigonometric Ratios of Some Standard Angles

Gk Scientist February 20, 2022 No Comments
Tweet WhatsApp Telegram

Table of Contents

  • Trigonometric Ratios of Some Standard Angles:
    • Trigonometric Ratios of π/4:
    • Trigonometric Ratios of π/6:
    • Trigonometric Ratios of π/3:
    • Trigonometric Ratios of 0 and π/2:

Trigonometric Ratios of Some Standard Angles:

In the case of two similar right-angled triangles, the ratios of the corresponding sides are the same. So, the trigonometric ratios with respect to equal acute angles are also the same irrespective of the lengths of the sides. Thus, for an acute angle of any magnitude, one can determine its trigonometrical ratios by drawing a right triangle with one of its angles equal to the given acute angle.

Trigonometric Ratios of π/4:

t ratios diagram - Trigonometric Ratios of Some Standard Angles

Let the revolving line OP starting from the initial position OX trace out ∠XOP = π/4 in the positive (anticlockwise) direction. From P, PN is drawn perpendicular to OX. Then, ∠NOP = π/4.

t ratios of pie by four - Trigonometric Ratios of Some Standard Angles

Trigonometric Ratios of π/6:

t ratios diagram of pie by six - Trigonometric Ratios of Some Standard Angles

Let ∠XOP = π/6. PN is drawn perpendicular to OX, and is produced upto P’ such that NP’ = PN.

Then, △PON ≅ △P’ON
∴ ∠P’ON = π/6, ∠OPN = π/3
Obviously, △POP’ is equilateral.
∴ OP = PP’ = r
⇒ PN = r/2
t ratios of pie by six - Trigonometric Ratios of Some Standard Angles

Trigonometric Ratios of π/3:

t ratios diagram of pie by 3 - Trigonometric Ratios of Some Standard Angles

Here, ∠XOP = π/3. PN is drawn perpendicular to OX.

Then, ∠OPN = π/6. ON is produced upto N’ such that ON = NN’.

Clearly, △ONP ≅ △N’NP
∴ ∠NN’P = π/3 and ∠N’PN = π/6
∴ △OPN’ is equilateral
∴ ON = 1/2 ON’ = 1/2 OP = 1/2 r
∴ PN2 = OP2 – ON2 = r2 – (r/2)2 = r2 – r2/4 = 3r2/4
trigonometric ratios of pie by three - Trigonometric Ratios of Some Standard Angles
t ratios of pie by three - Trigonometric Ratios of Some Standard Angles

Trigonometric Ratios of 0 and π/2:

t Ratios of 0 and pie by two - Trigonometric Ratios of Some Standard Angles

Let the revolving line OP, starting from OX, trace out a very small angle XOP = θ in the anticlockwise direction. From P, PN is drawn perpendicular to OX. If the point P is very close to N, then PN → 0, OP → ON and θ → 0 [Fig: (a)].

∴ sin 0 = PN/ON = 0/OP = 0
cos 0 = ON/OP = ON/ON = 1
tan 0 = PN/ON = 0/OP = 0

Obviously, cosec 0 = ON/PN = OP/0, undefined
sec 0 = 1
cot 0 = ON/PN = OP/0, undefined

Now, consider that the rotating line OP, starting from OX, trace out ∠XOP = θ, which is very nearly equal to π/2 in the anticlockwise direction.

Now, from P, PN is dropped perpendicular to OX. It is clear from [fig (b)] that as the angle θ approaches π/2, N approaches 0, PN → OP and ON → 0.

∴ sin π/2 = PN/OP = OP/OP = 1
cos π/2 = ON/OP = 0/OP = 0
tan π/2 = PN/ON = OP/0, undefined

Obviously, cosec π/2 = 1
sec π/2 = OP/ON = OP/0, undefined
and cot π/2 = ON/PN = 0/OP = 0

The table shows the values of trigonometric ratios for the standard angles 0, π/6, π/4, π/3 and π/2.

trigonometric ratios for the standard angles - Trigonometric Ratios of Some Standard Angles

A handy method of determining the values of the t-ratios for the standard angles is discussed below.

The first five whole numbers 0 to 4 are written below the circular measures 0, π/6, π/4, π/3 and π/2 respectively along the row for the values of sin θ. All these numbers are divided by 4 and then the positive square roots of the resulting numbers are taken as the values of sin θ. The values thus obtained for sin θ are then written in the reverse order to get the values of cos θ. Finally, the values of sin θ are divided by the corresponding values of cos θ to get the values of tan θ.

t ratios for the standard angles - Trigonometric Ratios of Some Standard Angles

Trigonometric FunctionsWerner Theory of Coordination Compounds
Kirchhoff’s LawsCrystal Field Theory or Ligand Field Theory
Wheat Stone Bridge PrincipleNuclear Reactor- Construction, Working and Uses
Transformer- Construction, Working and UsesElectrochemistry Notes From Tamil Board
Prev Article
Next Article

Related Articles

Cube Roots of Unity
Cube Roots of Unity: Let us denote the cube roots …
Gk Scientist March 22, 2022 Mathematics

Cube Roots of Unity

General Solution of an Equation of the Type (a cos θ + b sin θ = c)
General Solution of an Equation: One may attempt to solve …
Gk Scientist February 28, 2022 Mathematics

General Solution of an Equation of the Type (a cos θ + b sin θ = c)

Heights and Distances in Trigonometry
Heights and Distances in Trigonometry: Knowledge of Trigonometry can be …
Gk Scientist March 21, 2022 Mathematics

Heights and Distances in Trigonometry

Types of Relation
Types of Relation: (1) Identity Realtion- Relation R on a …
Gk Scientist September 26, 2022 Mathematics

Types of Relation

Formation of Quadratic Equations With Given Roots
Formation of Quadratic Equations With Given Roots: Let α and …
Gk Scientist May 9, 2022 Mathematics

Formation of Quadratic Equations With Given Roots

Projection Formulae and Area of a Triangle
Projection Formulae and Area of a Triangle: Projection Formulae: In …
Gk Scientist March 12, 2022 Mathematics

Projection Formulae and Area of a Triangle

Trigonometric Equations and their General Solutions
Trigonometric Equations and their General Solutions: It has been discussed …
Gk Scientist February 28, 2022 Mathematics

Trigonometric Equations and their General Solutions

Trigonometric Ratios of Associated Angles
Trigonometric Ratios of Associated Angles: You have already come to …
Gk Scientist February 21, 2022 Mathematics

Trigonometric Ratios of Associated Angles

Leave a Reply Cancel Reply

Search

  • Popular
  • Recent

GK SCIENTIST

General Knowledge One Stop Source

Information

  • About Us
  • Terms and Condition, Disclaimer
  • Contact Us

Android Apps

  • IAS App For English Medium Students
  • IAS Basics App For English Medium Students
  • IAS Hindi App For Hindi Medium Students
DMCA.com Protection Status

Popular Tags

Biology (34) Biology Questions (89) Chemistry (57) Computer (215) Current Affairs (4) Current Content (0) Economy (19) English Essay (172) English Grammar (75) English Literature (10) Geography (83) History (259) Indian Anthropology (11) Indian Polity (14) JKAS Mains Question Papers (17) Mathematics (68) Moral Science (7) NCERT & Other Boards Books (25) Philosophy (115) Physics (89) Political Science (134) RS TV News (33) Science (553) Social Anthropology (9) Social Science (17) Solved Paper (47) UPSC (7) UPSC Mains Question Papers (26)

Downloads

  • NCERT Books
  • Old NCERT Books
  • NIOS Books For IAS, SSC, and State PSC Exam
  • Tamil Nadu Board Books: Important For UPSC, SSC, and State PSC Exam
  • Modern Indian and World History Notes For IAS Exam
  • UPSC Topper 2013 Gaurav Agrawal Notes For IAS Preparation
  • UPSC IAS Prelims General Studies – Previous Year Papers
  • UPSC Mains Question Papers

Copyright © 2023 GK SCIENTIST
Theme by MyThemeShop.com& Hosted On Cloudways

Ad Blocker Detected

Our website is made possible by displaying online advertisements to our visitors. Please consider supporting us by disabling your ad blocker.

Refresh
 

Loading Comments...