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Trigonometry Basics

by Satish

Definition of Trigonometry: Trigonometry considers the properties of angles and certain ratios associated with angles, and applies the knowledge of these properties to the solution of triangles and various other algebraic and geometric problems. Incidentally trigonometry considers also certain time-saving aids in computation such as logarithms, which are generally employed in the solution of triangles. Briefly stated,

Trigonometry is the science of angular magnitudes and the art of applying the principles of this science to the solution of problems.

The word Trigonometry comes from two Greek words, trigonon = triangle, and metron = measure. The method was originated in the second century B.C. by Hipparchus and other early Greek astronomers in their attempts to solve certain spherical triangles. The term trigonometry was not used until the close of the sixteenth century.

Before we get into the basic definitions of Trigonometric Functions, let us look at the basic definition of a function_.

Definition of Function: When two variables are so related that the value of the one depends upon the value of the other, the one is said to be a function of the other.

EXAMPLES: The area of a square is a function of its side.
The volume of a sphere is a function of its radius.
The velocity of a falling body is a function of the time elapsed since it began to fall.
The output of a factory is a function of the number of men employed.
In the expression y=\frac{x-1}{x+1}, y depends upon x for its value, hence y is a function of x.

Definition of Reciprocal: If the product of two quantities equals unity, each is said to be the reciprocal of the other.

For example, if xy = 1, x is the reciprocal of y, and y is the reciprocalof x.
1/2 is the reciprocal of 2, and 2 is the reciprocal of 1/2, for 1/2X2=1.
In general, a/b and b/a are reciprocals since a/bxb/a=1 .
From xy = 1 it follows that x = 1/y, and y = 1/x, that is,
_The reciprocal of any quantity is unity divided by that quantity.

Six Trigonometric Functions of an Acute Angle: Let A be any acute angle, B any point on either side of the angle, and ABC the right triangle formed by drawing a perpendicular from B to the other side of the angle. Denote AC, the side adjacent to the angle A, by b (for base), BC, the side opposite the angle A, by a (for altitude), and the hypotenuse AB by h.

The three sides of the right triangle form six different ratios, namely,
\frac{a}{h},\frac{b}{h},\frac{a}{b},
and their reciprocals
\frac{h}{a},\frac{h}{b},\frac{b}{a},
|{border-color:white}.Photo 17276

Since these ratios depend upon the angle for their values, they are the functions of the angle according to the general definition of a function that we discussed at the beginning of our lesson. Each of these functions has received a special name.

\frac{a}{h} \quad i.e \quad \frac{side \quad opposite\quad angle \quad A}{hypotenuse}

is called Sine of angle A

\frac{b}{h} \quad i.e \quad \frac{side \quad adjacent\quad angle \quad A}{hypotenuse}

is called Cosine of angle A

\frac{a}{b} \quad i.e \quad \frac{side \quad opposite\quad angle \quad A}{side \quad adjacent\quad angle \quad A}

is called Tangent of angle A

\frac{h}{a}

i.e the reciprocal of the Sine is called Cosecant of angle A

\frac{h}{b}

i.e the reciprocal of the Cosine is called Secant of angle A

\frac{b}{a}

i.e the reciprocal of the Tangent is called Cotangent of angle A

The six functions just defined are variously known as the trigonometric, circular, or goniometric functions: trigonometric, because they form the basis of the science of trigonometry; circular, because of their relations to the arc of a circle; goniometric, because of their use in determining angles, from gonia, a Greek word meaning angle.

The terms sine of angle A, cosine of angle A, etc., are abbreviated to sin A, cos A, tan A, cosec A , sec A , and cot A. The definitions of the first six trigonometric functions must be thoroughly memorized. The first three are especially important and should be memorized.The remaining three functions may be remembered most readily by the aid of the reciprocal relations, reciprocal relations,

Sin A.Cosec A = 1
Cos A.Sec A = 1
Tan A.Cot A=1


It should be noticed that while a, b, and h are lines, the ratio of any two of them is an abstract number; that is, the trigonometric functions are abstract numbers. Also, the expressions sin A cos A, tan A etc., are single symbols which cannot be separated, sin has no meaning except as it is associated with some angle.

EXAMPLE: The sides of a right triangle are 3, 4, 5. Find all the trigonometric functions of the angle A opposite the side 4.

Solution: The hypotenuse of the triangle equals 5. Hence, applying the definitions, we have

SinA = \frac{BC}{AB}=\frac{4}{5}
CosA = \frac{AC}{AB}=\frac{3}{5}
TanA = \frac{BC}{AC}=\frac{4}{3}
CosecA = \frac{1}{SinA}=\frac{5}{4}
SecA = \frac{1}{CosA}=\frac{5}{3}
CotA = \frac{1}{TanA}=\frac{3}{4}
Photo 17282


Basic Identities

Sinx=\frac{1}{Cosecx}  Cosx = \frac{1}{Secx} Tanx=\frac{1}{Cotx}
 Cosecx = \frac{1}{Sinx}  Secx = \frac{1}{Cosx}  Cotx= \frac{1}{Tanx}
Tanx=\frac{Sinx}{Cosx}  Cotx =\frac{Cosx}{Sinx}
Phythagorean Identities Symmetry Properties
 Sin^2x + Cos^2x = 1
 Sec^2x - Tan^2x = 1
 Cosec^2x - Cot^2x = 1
 Cos(-x) = Cosx \quad Sec(-x) = Secx
Sin(-x) = -Sinx \quad Cosce(-x) = -Cosecx
 Tan(-x) = -Tanx \quad Cot(-x) = -Cotx



Graphs of the Six Trigonometric Functions

Sinx Cosx
Tanx Cotx
Secx Cosec

Image Credit: robindegrassi

27 Comments
    bandongri
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    bandongriSat, 23 Jun 2012 08:02:08 -0000

    really a good job it is the correct basic to start from

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    bibhas2012
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    bibhas2012Sat, 21 Apr 2012 17:16:06 -0000

    Very useful

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    raheza
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    rahezaSat, 16 Jul 2011 15:09:22 -0000

    2/10=2 ,challenge

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    uddeshya
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    uddeshyaThu, 12 May 2011 04:01:48 -0000

    realllyyyy ossum!
    i was alien to trigonometry n this helped me alot to get sum knowledge
    well now can u plzzzzzzzz increase d level of toughness!

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    neesu
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    neesuWed, 15 Dec 2010 15:24:28 -0000

    A+B=10 B+C=10 C+D=10 D+A=10 find the value of A,B,C,D Challenge

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    ashishhbti
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    ashishhbtiFri, 23 Mar 2012 19:15:54 -0000

    ans is A=5 B=5 c=5 D=5

    I_Universe
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    I_UniverseTue, 17 Apr 2012 11:28:47 -0000
    lavanesh89
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    Wed, 20 Jun 2012 06:20:15 -0000

    a=5,b=5,c=5,d=5

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    Chandrakanth Nawgrahe
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    Chandrakanth NawgraheThu, 12 Aug 2010 10:16:42 -0000

    thanks for adequate information
    learn more,will in continue.

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    wildgoose
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    wildgooseWed, 30 Jun 2010 11:37:42 -0000

    I didn't know basics of trigonometry. This was greatly useful!!!!

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    ramavtar_verma
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    Ram Avtar VermaMon, 18 Jan 2010 10:25:09 -0000

    sin (A+B)= sin a cos b + cos a sin b

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    avi_geom
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    avinashMon, 11 Jan 2010 19:26:16 -0000

    the best ways to remember---

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    saber_ali
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    SaberThu, 19 Nov 2009 16:33:07 -0000

    Hi dears
    Will you any one solve this problem that how can i get the length of the hyportenious when i know the on angle (90" andgle) and the both side lenths of the triangle

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    nikhilpalakkal
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    nikhilpalakkalMon, 25 Jan 2010 14:08:40 -0000

    thanks.really useful

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    bhattavantika
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    avantikabhattWed, 18 Nov 2009 13:21:25 -0000

    quite good basic

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    aayushee001iit
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    aayushee jindalFri, 02 Oct 2009 09:00:49 -0000

    thnxx…….dese wer really vry gud

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    Unique Gene
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    Koushik VikramTue, 11 Aug 2009 15:02:37 -0000

    Good.These are very basic concepts.Yeah all of u r right.But we need not complain.IIT-JEE wants your basic concepts to be very strong.

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    userdce
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    user dceSat, 30 May 2009 07:53:21 -0000

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    Sureshbala
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    kvsrharsha
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    kvsrharshaWed, 29 Apr 2009 09:44:28 -0000

    very good but need some more

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    jackcamer
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    Jack camerSat, 25 Apr 2009 15:44:03 -0000

    when can we expect the advanced topics…

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    abhinayah
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    abhinayahSat, 25 Apr 2009 06:22:44 -0000

    good basics brush up

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    deepakkumar753
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    deepak agarwalThu, 19 Mar 2009 10:36:53 -0000

    really a good approch.thanx to author

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    coolvenky9
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    venkyTue, 03 Feb 2009 09:47:12 -0000

    i think this is more than enough for GMAT exam . can anyone tell am i correct?

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    oLahav
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    Oren LahavTue, 03 Feb 2009 14:55:26 -0000

    Yes, I agree that this is more than enough for the GMAT. The GMAT does not tend to put high emphasis on trigonometry, identities and invertible functions. The emphasis is mostly on arithmetic and algebra, with some geometry and statistics thrown in.

    For other tests, however, this may be very helpful. Besides, it's always good to be prepared and know as much as you can, but it may be more beneficial to concentrate on the topics that appear on the GMAT more often.

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    Sureshbala
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    SureshFri, 16 Jan 2009 17:41:05 -0000

    Dear Bharuchi,

    Please give us some time…very soon we are going to come up with a series of lessons and tests on the advanced concepts of Trigonometry

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    bharuchi
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    bharuchiMon, 29 Dec 2008 09:15:59 -0000

    actually very basic.
    V need something more.

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    polampalli
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    aparanjaniSun, 16 Nov 2008 09:05:10 -0000

    very basic good

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    Leonardodvin
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    SatishWed, 29 Oct 2008 19:13:02 -0000

    Hi…this is very basic lesson and we will definitely have a series of lessons where we will discuss advanced concepts as well.

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    samir_agrawal
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    samir_agrawalWed, 29 Oct 2008 18:56:51 -0000

    very basic everyone knows this i thought it would be something new and tough.

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    pandurang68
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    pandurang68Wed, 29 Oct 2008 16:02:37 -0000

    excellent really

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    shivam07usit
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    shivam07usitWed, 29 Oct 2008 15:08:19 -0000

    good but very basic

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Last Updated At Dec 13, 2012
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