Trigonometry

TrigonometryHypotenuse Adjacent Opposite NAME Karan CLASS 10TH CROLL NO. - 9Trigonometry The word trigonometry is derived from the Greek words tri(meaning three), gon (meaning sides) and metron (meaning measure).

Trigonometry is the study of relationships between the sides and angles of a triangle.

Early astronomers used it to find out the distances of the stars and planets from the Earth.

Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrical concepts. iTutor. 2000-2013. All Rights ReservedRight TriangleA triangle in which one angle is equal to 90 is called right triangle.

The side opposite to the right angle is known as hypotenuse.AC is the hypotenuseThe other two sides are known as legs. AB and BC are the legsTrigonometry deals with Right TrianglesACBHypotenuse iTutor. 2000-2013. All Rights ReservedPythagoras TheoremIn any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs. ACBHypotenuse (Hypotenuse)2 = (Perpendicular)2 + (Base)2AC2 = BC2 + AB2 iTutor. 2000-2013. All Rights ReservedPythagoras TheoremPythagoras Theorem Proof:Given: ABC is a right angled triangle where B = 900 And AB = P, BC= b and AC = h.To Prove: h2 = p2 + b2Construction : Draw a BD from B to AC , where AD = x and CB = h-x , Proof : In ABC and ABD, ABC ABD --------(AA)In ABC and BDC both are similar So by these similarity,

pbhABCDx(h-x)5Pythagoras TheoremOr P2 = x h And b2 = h (h x) Adding both L.H.S. and R.H. S. Thenp2 + b2 = (x h) + h (h x)Or p2 + b2 = xh + h2 hx Hence the Pythagoras theorem p2 + b2 = h2

pbhABCDx(h-x)Trigonometric RatiosLet us take a right triangle ABC

Here, ACB () is an acute angle.

The position of the side AB with respect to angle . We call it the side opposite to angle .

AC is the hypotenuse of the right triangle and the side BC is a part of . So, we call it the side adjacent to angle .ACBHypotenuse Side opposite to angle Side adjacent to angle iTutor. 2000-2013. All Rights Reserved7Trigonometric RatiosThe trigonometric ratios of the angle C in right ABC as follows :Sine of C =

=

Cosine of C=

= ACBHypotenuse Side opposite to angle Side adjacent to angle Side opposite to CHypotenuse ABACSide adjacent to CHypotenuse BCAC iTutor. 2000-2013. All Rights ReservedTrigonometric Ratios Tangent of C =

=

Cosecant of C=

=

Secant of C = ACBHypotenuse Side opposite to angle Side adjacent to angle Side opposite to CSide adjacent to CABBCSide adjacent to CHypotenuse Side opposite to CHypotenuse ACABACAB= iTutor. 2000-2013. All Rights ReservedTrigonometric RatiosCotangent of C

Above Trigonometric Ratio arbitrates as sin C, cos C, tan C , cosec C , sec C, Cot C . If the measure of angle C is then the ratios are :sin , cos , tan , cosec , sec and cot ACBHypotenuse Side opposite to angle Side adjacent to angle Side opposite to CSide adjacent to CABBC== iTutor. 2000-2013. All Rights ReservedRelation among T-ratio Tan = Cosec = 1 / Sin

Sec = 1 / Cos

Cot = Cos / Sin = 1 / Tan ACBpbh iTutor. 2000-2013. All Rights Reserved

11TableSin = p / h Cos = b / hTan = p / bCosec = h / pSec = h / bCot = b / p

ACBpbh iTutor. 2000-2013. All Rights ReservedValues of Trigonometric functionT. Ratios 030456090Sine0 1/23/21Cosine13/21/2 0Tangent01/ 313Not definedCosecantNot defined222/ 31Secant12/ 322Not definedCotangentNot defined311/ 30 iTutor. 2000-2013. All Rights Reserved13Relation of angles with T. RatiosRelation of with Sin when 00 900 The greater the value of , the greater is the value of Sin.Smallest value of Sin = 0Greatest value of Sin = 1Relation of with Cos when 00 900 The greater the value of , the smaller is the value of Cos.Smallest value of Cos = 0Greatest value of Cos = 1

iTutor. 2000-2013. All Rights ReservedRelation of angles with T. RatiosRelation of with tan when 00 900 Tan increases as increases But ,tan is not defined at = 900 Smallest value of tan = 0 iTutor. 2000-2013. All Rights ReservedSome More Formulae If 00 900 Sin(900- ) = Cos Cos(900- ) = Sin

If 00< 900Tan(900- ) = Cot Sec(900- ) = Cosec

If 00 < 900Cot(900- )= Tan Cosec(900- ) = Sec

ACBpbh iTutor. 2000-2013. All Rights ReservedSome More Formulae Sin2 +Cos2 = 1

Sec2 -Tan2 = 1

Cosec2 - Cot2 = 1

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