BY SIBY SEBASTIAN PGT(MATHS) TRIGONOMETRIC FUNCTIONS BASIC IDEAS BY SIBY SEBASTIAN PGT(MATHS) siby sebastian pgt maths

Basic Terms The rotation of the terminal side of an angle counterclockwise. The rotation of the terminal side is clockwise. A C B siby sebastian pgt maths

Angle Measures and Types of Angles The most common unit for measuring angles is the degree. (One rotation = 360o) ¼ rotation = 90o, ½ rotation = 180o, 1 360 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛= 1 0 Angle and measure of angle are not the same, but it is common to say that an angle = its measure Types of angles named on basis of measure: siby sebastian pgt maths

Measuring Angles So far we have measured angles in degrees For most practical applications of trigonometry this is the preferred measure For advanced mathematics courses it is more common to measure angles in units called “radian measure” siby sebastian pgt maths

siby sebastian pgt maths Radian Measure An angle with its vertex at the center of a circle of radius ‘r’ units subtended by an arc of length ‘r’ unit is 1 radian. (1 rad) siby sebastian pgt maths

Comments on Radian Measure Since a complete rotation of a ray back to the initial position generates a circle of radius “r”, and the circumference of that circle (arc length) is 2𝝅𝒓, there are 2𝝅 radians in a complete rotation Based on the reasoning just discussed: 2𝝅 rad = 3600 , 𝝅 rad = 1800 1 rad = 𝟏𝟖𝟎 𝟎 𝝅 ≈ 𝟓𝟕.𝟑 𝟎 𝟏 𝟎 = 𝝅 𝒓𝒂𝒅 𝟏𝟖𝟎 siby sebastian pgt maths

Conversion Between Degrees and Radians Multiply a degree measure by 𝝅 𝟏𝟖𝟎 and simplify to convert to radians. Multiply a radian measure by 𝟏𝟖𝟎 𝝅 and simplify to convert to degrees. siby sebastian pgt maths

Convert Degrees to Radians 𝟔𝟎 𝟎 = 𝟔𝟎 𝟎 𝒙 𝝅 𝟏𝟖𝟎 𝟎 𝒓𝒂𝒅𝒊𝒂𝒏= 𝝅 𝟑 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 b) 221.7 221.70 = 𝟐𝟐𝟏.𝟕 𝟎 𝒙 𝝅 𝟏𝟖𝟎 𝟎 𝒓𝒂𝒅≈𝟑.𝟖𝟗𝟔 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 siby sebastian pgt maths

Convert Radians to Degrees 𝟏𝟏𝝅 𝟒 𝒓𝒂𝒅 = 𝟏𝟏𝝅 𝟒 𝒓𝒂𝒅 x 𝟏𝟖𝟎 𝟎 𝝅 𝒓𝒂𝒅 = 𝟒𝟗𝟓 𝟎 b) 3.25 rad 3.25 rad = 𝟑.𝟐𝟓 𝒓𝒂𝒅 𝟏 x 𝟏𝟖𝟎 𝟎 𝝅 𝒓𝒂𝒅 ≈ 𝟏𝟖𝟔.𝟐 𝟎 siby sebastian pgt maths

Equivalent Angles in Degrees and Radians siby sebastian pgt maths

Trigonometric Functions In a circle of radius ‘r’ units and if P(x,y) is a point on the circle then the trigonometric functions are defined by sin𝜽= 𝒚 𝒓 cosec𝜽= 𝒓 𝒚 cos𝜽= 𝒙 𝒓 sec𝜽= 𝒓 𝒙 tan𝜽= 𝒙 𝒚 cot = 𝒚 𝒙 r y x siby sebastian pgt maths

Trigonometric Functions “Circular Functions” are named as trig functions (sine, cosine, tangent, etc.) The domain of trig functions is a set of angles measured either in degrees or radians The domain of circular functions is the set of real numbers The value of a trig function of a specific angle in its domain is a ratio of real numbers The value of circular function of a real number “x” is the same as the corresponding trig function of “x radians” siby sebastian pgt maths

Exponential Notation and Trigonometric Functions sin2 A = (sin A)2 tan3A = (tanA)3 Sec5A = (secA)5 siby sebastian pgt maths

Signs of Trig Functions by Quadrant of Angle Considering the following three functions and the sign of x, y and r in each quadrant, which functions are positive in each quadrant? siby sebastian pgt maths

“All students take calculus” Mnemonic Techniques It will help to memorize by learning these words in Quadrants I - IV: “All students take calculus” And remembering reciprocal identities Trig functions are negative in quadrants where they are not positive siby sebastian pgt maths

Domain and Range of Sine Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sin A = y/r Domain of sine function is the set of all A for which y/r is a real number. Since r can’t be zero, y/r is always a real number and domain is “any angle” Range of sine function is the set of all y/r, but since y is less than or equal to r, this ratio will always be equal to 1 or will be a proper fraction, positive or negative: siby sebastian pgt maths

siby sebastian pgt maths GRAPH OF sine FUNCTION Click here to see how sin function is generated siby sebastian pgt maths

Domain and Range of Cosine Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cos A = x/r Domain of cosine function is the set of all A for which x/r is a real number. Since r can’t be zero, x/r is always a real number and domain is “any angle” Range of cosine function is the set of all x/r, but since x is less than or equal to r, this ratio will always be equal to 1, -1 or will be a proper fraction, positive or negative: siby sebastian pgt maths

GRAPH OF cosine FUNCTION Click here to see how cosine function is generated siby sebastian pgt maths

Domain and Range of Tangent Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, tan A = y/x Domain of tangent function is the set of all A for which y/x is a real number. Tangent will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o Range of tangent function is the set of all y/x, but since all of these are possible: x=y, x<y, x>y, this ratio can be any positive or negative real number: siby sebastian pgt maths

GRAPH OF tangent FUNCTION Click here to see how tangent function is generated siby sebastian pgt maths

Domain and Range of Cosecant Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, csc A = r/y Domain of cosecant function is the set of all A for which r/y is a real number. Cosecant will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o Range of cosecant function is the reciprocal of the range of the sine function. Reciprocals of numbers between -1 and 1 are: siby sebastian pgt maths

GRAPH OF cosecant FUNCTION siby sebastian pgt maths

Domain and Range of Secant Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sec A = r/x Domain of secant function is the set of all A for which r/x is a real number. Secant will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o Range of secant function is the reciprocal of the range of the cosine function. Reciprocals of numbers between -1 and 1 are: siby sebastian pgt maths

GRAPH OF secant FUNCTION siby sebastian pgt maths

Domain and Range of Cotangent Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cot A = x/y Domain of cotangent function is the set of all A for which x/y is a real number. Cotangent will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o Range of cotangent function is the reciprocal of the range of the tangent function. The reciprocal of the set of numbers between negative infinity and positive infinity is: siby sebastian pgt maths

GRAPH OF cotangent FUNCTION siby sebastian pgt maths

Ranges of Trigonometric Functions For any angle  for which the indicated functions exist: 1  sin   1 1  cos   1 tan  and cot  can take any real number sec   1 or sec   1 csc   1 or csc   1. Note that sec  and csc  are never between 1 and 1 siby sebastian pgt maths

siby sebastian pgt maths Periodic Properties siby sebastian pgt maths

Theorem Even-Odd Properties siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 1.sin( 𝝅 𝟐 −𝒙)=𝒄𝒐𝒔𝒙 2.cos( 𝝅 𝟐 −𝒙) = sinx 3.tan( 𝝅 𝟐 −𝒙) = cotx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 4.sin( 𝝅 𝟐 +𝒙)=𝒄𝒐𝒔𝒙 5.cos( 𝝅 𝟐 +𝒙) = - sinx 6.tan( 𝝅 𝟐 +𝒙) = - cotx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 7.sin(𝛑−𝒙)=𝐬𝐢𝐧𝐱 8.cos(𝛑−𝒙) = -cosx 9.tan(𝛑 −𝒙) = - tanx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 10.sin(𝛑+𝒙)=−𝐬𝐢𝐧𝐱 11.cos(𝛑+𝒙) = -cosx 12.tan(𝛑+𝒙) = tanx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 13.sin( 𝟑𝝅 𝟐 −𝒙)=−𝒄𝒐𝒔𝒙 14.cos( 𝟑𝝅 𝟐 −𝒙) = -sinx 15.tan( 𝟑𝝅 𝟐 −𝒙) = cotx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 16. sin( 𝟑𝝅 𝟐 +𝒙)=−𝒄𝒐𝒔𝒙 17 .cos( 𝟑𝝅 𝟐 +𝒙) = sinx 18 .tan( 𝟑𝝅 𝟐 +𝒙) = - cotx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 19.sin(𝟐𝛑−𝒙)=−𝐬𝐢𝐧𝐱 20.cos(𝟐𝛑−𝒙) = cosx 21.tan(2𝛑 −𝒙) =-tanx siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS sin2x +cos2x =1 1+tan2x =sec2x 1+cot2x =cosec2x siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS SUM AND DIFFERENCE OF TWO ANGLES 1.cos(x + y) = cosxcosy – sinxsiny 2.cos(x – y) = cosxcosy + sinxsiny 3.sin(x + y) = sinxcosy + cosxsiny 4.sin( x – y) = sinxcosy - cosxsiny siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 5.tan(x + y) = 𝒕𝒂𝒏𝒙+𝒕𝒂𝒏𝒚 𝟏−𝒕𝒂𝒏𝒙𝒕𝒂𝒏𝒚 6.tan(x – y) = 𝒕𝒂𝒏𝒙−𝒕𝒂𝒏𝒚 𝟏+𝒕𝒂𝒏𝒙𝒕𝒂𝒏𝒚 7.cot(x + y) = 𝒄𝒐𝒕𝒙𝒄𝒐𝒕𝒚 −𝟏 𝒄𝒐𝒕𝒚+𝒄𝒐𝒕𝒙 8.cot(x - y) = 𝒄𝒐𝒕𝒙𝒄𝒐𝒕𝒚+𝟏 𝒄𝒐𝒕𝒚−𝒄𝒐𝒕𝒙 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS PRODUCT AS SUM OR DIFFERENCE 1 .2sinxcosy = sin(x + y) + sin(x – y) 2. 2cosxsiny = sin(x + y) – sin(x – y) 3.2cosxcosy = cos(x + y)+cos(x – y) 4.-2sinxsiny = cos(x + y) – cos(x – y) siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS SUM OR DIFFERENCE AS PRODUCT 1.sinx + siny = 2sin( 𝒙+𝒚 𝟐 )𝒄𝒐𝒔( 𝒙−𝒚 𝟐 ) 2.sinx – siny = 2cos( 𝒙+𝒚 𝟐 )𝒔𝒊𝒏( 𝒙−𝒚 𝟐 ) 3.cosx + cosy = 2cos( 𝒙+𝒚 𝟐 )𝒄𝒐𝒔( 𝒙−𝒚 𝟐 ) 4.cosx – cosy = - 2sin( 𝒙+𝒚 𝟐 )𝒔𝒊𝒏( 𝒙−𝒚 𝟐 ) siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS MULTIPLE ANGLES 1.sin2x = 2sinxcosx = 𝟐𝒕𝒂𝒏𝒙 𝟏+ 𝒕𝒂𝒏 𝟐 𝒙 2.cos2x = cos2x – sin2x = 2cos2x – 1 = 1 – 2sin2x = 𝟏− 𝒕𝒂𝒏 𝟐 𝒙 𝟏+ 𝒕𝒂𝒏 𝟐 𝒙 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 3.tan2x = 𝟐𝒕𝒂𝒏𝒙 𝟏− 𝒕𝒂𝒏 𝟐 𝒙 4.sin3x = 3sinx – 4sin3x 5.cos3x = 4cos3x – 3cosx 6.tan3x = 𝟑𝒕𝒂𝒏𝒙 − 𝒕𝒂𝒏 𝟑 𝒙 𝟏−𝟑 𝒕𝒂𝒏 𝟐 𝒙 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS SUB MULTIPLE ANGLES 1.sinx = 2sin 𝒙 𝟐 𝒄𝒐𝒔 𝒙 𝟐 2.cosx = 𝒄𝒐𝒔 𝟐 𝒙 𝟐 − 𝒔𝒊𝒏 𝟐 𝒙 𝟐 3.1- cosx = 2 𝒔𝒊𝒏 𝟐 𝒙 𝟐 4.1+cosx = 2 𝒄𝒐𝒔 𝟐 𝒙 𝟐 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS GENERAL SOLUTIONS 1.sinx =0 then x= n𝝅, n∈𝒁 2.cosx = 0 then x=(2n + 1) 𝝅 𝟐 , n∈𝒁 3.tanx =0 then x= n𝝅, n∈𝒁 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS 4.Sinx = siny then,x = n𝝅+ (−𝟏) 𝒏 𝒚, n∈𝒁 5.cosx =cosy then, 𝒙=𝟐𝒏𝛑±y ,n∈𝒁 6.tanx = tany then x= n𝝅+𝒚, n∈𝒁 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS Sine Rule 𝒂 𝒔𝒊𝒏𝑨 = 𝒃 𝒔𝒊𝒏𝑩 = 𝒄 𝒔𝒊𝒏𝑪 siby sebastian pgt maths

BASIC RULES OF TRIGONOMETRIC FUNCTIONS Cosine Rule cosA = 𝒃 𝟐 + 𝒄 𝟐 − 𝒂 𝟐 𝟐𝒃𝒄 cosB = 𝒄 𝟐 +𝒂− 𝒃 𝟐 𝟐𝒄𝒂 cosC = 𝒂 𝟐 + 𝒃 𝟐 − 𝒄 𝟐 𝟐𝒂𝒃 siby sebastian pgt maths

Finally let us dance together and enjoy trigonometry Practice & Until you get it. …….. siby sebastian pgt maths