Trigonometric Functions Study and Definitions in Mathematics
In this educational material, explore the study of trigonometric functions in mathematics, covering definitions, important points, values, and applications. Learn about trigonometric ratios, unit circle representation, and key concepts related to sine, cosine, tangent, secant, cosecant, and cotangent functions. Understand how these functions are applied in various mathematical problems and real-world scenarios.
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an International CBSE Finger Print School Coimbatore SUBJECT NAME - 041 MATHEMATICS GRADE-XI UNIT 3 TOPIC TRIGONOMETRIC FUNCTIONS TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY 12/06/23 1
TRIGONOMETRIC FUNCTIONS In lower class, we have studied the Trigonometric ratios using the Right angle Triangle. In a Right angled triangle, the side opposite to right angle is known as Hypotenuse and other two sides are opposite side and adjacent side which are relative terms i.e they depend on the angle chosen. The ratio of Opposite side to the given angle A to Hypotenuse is defined as SINE ratio of angle A and is denoted with SinA. Similarly other ratios are defined as follows. 12/06/23 2/17 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
DEFINITIONS Now we are going to introduce the TRIGONOMETRIC RATIOS as funtions independent of right angled triangle in this 11thclass using a unit circle. In Higher classes you will be studying Trigonometric functions independent of any geometrical figure. Let us consider a unit circle centred at Origin i.e a circle with radius one unit and centre at Origin (0,0). (Note: We may also chose a circle with radius a units). Let P = (a,b) be any point on the circle such that XOP = x radians. Then OP = 1 as the distance between centre and any point on the circle is 1. We define the Trigonometric functions as follows. 12/06/23 3/17 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
DEFINITIONS IMPORTANT POINTS The above definitions can be applied to any angle (not only to acute angle). Tan x is not defined for a = 0 and cot x is not defined for b = 0. Similarly Sec x is not defined for a = 0 and Cosec x is not defined for b = 0. That is Tanx and Secx are not defined for odd multiples of 90 degrees and Cotx and Cosecx are not defined for integral multiples of 180 degrees Sin x = b Cos x = a Tan x = b/a Cosec x = 1/b Sec x = 1/a Cot x = a/b 12/06/23 4/17 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
VALUES OF TRIGONOMETRIC FUNCTIONS If the point P = (a,b) lies on +ve Y-axis then we have a=0 and b=1 and x = /2 radians. Therefore Sin( /2) = 1 Cos( /2) = 0 Csc( /2) = 1 Sec( /2) = Not defined Tan( /2) = Not defined Cot( /2) = 0 If the point P is on +ve X- axis i.e on OX, then a=1 , b=0 and x = 0 radians. Therefore Sin0 = 0 Cos0 = 1 Tan0 = 0 Sec0 = 1 Csc0 = Not defined Cot0 = Not defined If the point P=(a,b) lies on ve Y-axis then we have a=0 , b=-1 and x = 3 /2 radians. So Sin(3 /2) = -1 Cos(3 /2) = 0 Tan(3 /2) = Nd Cot(3 /2) = 0 Sec(3 /2) = Nd Csc(3 /2) = -1 12/06/23 5/17 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
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Sign of Trigonometric Functions The Quadrant in which the angle lies is determined by the position of the terminal side of the angle. If the angle lies in Q1, then for the Point P = (a,b) both coordinates are +ve and hence all the Trigonometric ratios are positive. If the angle lies in Q2, then a is ve and b is +ve and hence Sin and Csc are +ve and rest are ve. If the angle lies in Q3, then a is ve and b is ve and hence Tan and Cot are +ve and rest are ve. If the angle lies in Q4, then a is +ve and b is ve and hence Cos and Sec are +ve and rest are ve. 12/06/23 7/17 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
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VALUES OF X FOR SIN & COS From the figure it is clear that Sinx = b and Cosx = a Sin(-x) = -b and Cos(-x) = a Hence Sin(-x) = - Sinx And Cos(-x) = Cosx 9/17 12/06/23 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
DOMAIN AND RANGES ORIENTATION 12/06/23 10/17 10 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
DERIVATION OF FORMULA (1)Cos(x+y) = CosxCosy SinxSiny for x,y R We prove the above result using the unit circle. Consider a Unit circle with centre at origin and angles as shown in figure. 11/17 12/06/23 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
DERIVATION The Triangles P1OP3 and P2OP$are congruent by SAS congruence. Therefore P1P3and P2P4 are equal. P1P32= [ Cosx Cos(-y)]2+ [ Sinx Sin(-y)]2 =[ Cosx Cosy]2 + [Sinx + Siny]2 = Cos2x + Cos2y 2CosxCosy + Sin2x + Sin2y + 2SinxSiny = 2 2[CosxCosy SinxSiny].........................(1) P2P42= [ 1 Cos(x+y)]2+ [0 Sin(x+y)]2 = 1 2Cos(x+y) +Cos2(x+y) + Sin2(x+y) = 2 2Cos(x+y) ................................(2) From (1) and (2), Cos(x+y) = CosxCosy - SinxSiny 12/06/23 12/17 12 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
SUM & PRODUCT TRANSFORMATIONS 12/06/23 13/17 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
TRIGONOMETRIC EQUATIONS Equations involving trigonometric functions of a variable are called Trigonometric equations. Eg. Sinx = , Cos2x = 1 , Sinx+ Cosx = 0 etc., The values of Sinx and Cosx repeat after an interval of 2 , and Tanx values repeat after an interval of , the Trigonometric equations have infinite no of solutions. For example Cosx = 1 is satisfied by x = 2n where n is Integer. PRINCIPAL & GENERAL SOLUTIONS The solutions of a Trigonometric equation for which 0 x<2 are called Principal solutions. The expression involving integer n which gives all solutions of a Trigonometric equation is called the General solution. For example Principal Solution of the equation Sinx = is x = /6 and x = 5 /6 but there are infinitely many values of x satisfied by the equation which comes under General solution. 12/06/23 14/17 14 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
Types of equations Trigonometric equations can be solved either by factorisation, using sum and product transformations or by using quadratic formula. If the equation is of the form aCosx + b Sinx = c then divide both sides with Now we can convert it into the form Sin(x+y) or Cos(x+y) Then it can be solved. SUPPLIMENTARY READING In Traingle ABC, the sides BC,CA,AB are respectively denoted by a , b and c respectively and the internal angles opposite to sides are represented by A,B,and C. 15/17 12/06/23 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
Questions: 1. Convert 6 radians into degree measure. 2. If in two circles, arcs of the same length subtend angles 60 and 75 at the centre, find the ratio of their radii. 3. If cot x = 5/12 , x lies in second quadrant, find the values of other five trigonometric functions. 4. Prove that 12/06/23 16/17 16 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
References https://vdocuments.mx/ppt-on-trigonometric-functions-class-11.html https://www.slideshare.net/RushikeshReddy1998/ppt-on-trigonometric- functions-class-11 https://ux1.eiu.edu/~aalvarado2/TRIGONOMETRY.ppt https://ncert.nic.in/pdf/publication/exemplarproblem/classXI/mathematic s/keep203.pdf https://www.mathongo.com/ncert-solutions/ncert-solutions-class-11- maths-chapter-3 17/17 12/06/23 TRIGONOMETRIC FUNCTIONS/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY
