basic rules of trigonometric functions
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BY SIBY SEBASTIAN PGT(MATHS) siby sebastian pgt maths A C B The rotation of the terminal side of an angle counterclockwise. The rotation of the terminal side is clockwise. siby sebastian pgt maths The most common unit for measuring angles is the degree. (One rotation = 360o) ¼ rotation = 90o, ½ rotation = 180o, 1 πππ‘ππ‘πππ 360 = 10 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 siby sebastian pgt maths 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 Based on the reasoning just discussed: 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 2π π, there are 2π radians in a complete rotation 2π rad = 3600 , π rad = 1800 1 rad ππππ = π ππ = siby sebastian pgt maths ≈ ππ. ππ π πππ πππ Multiply a degree measure by Multiply a radian measure by and simplify to convert to radians. and simplify to convert to degrees. π πππ πππ π siby sebastian pgt maths a) 60ο° π π ππ = ππ π πππ πππ = πππ ππππ π πππ π π π b) 221.7 221.70 = πππ. ππ π π πππ ππππ ≈ π. πππ πππ ππππ siby sebastian pgt maths πππ a) π πππ π πππ πππ π πππ = πππ x ππππ π πππ π = πππ b) 3.25 rad 3.25 rad = π.ππ πππ ππππ x π π πππ ≈ πππ. ππ siby sebastian pgt maths siby sebastian pgt maths r 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 π π sinπ½ = cosecπ½ = y π x siby sebastian pgt maths π cosπ½ = π π secπ½ = tanπ½ = π π π π cot = π π “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 siby sebastian pgt maths The value of circular function of a real number “x” is the same as the corresponding trig function of “x radians” • sin2 A = (sin A)2 • tan3A = (tanA)3 • Sec5A = (secA)5 siby sebastian pgt maths • 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 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 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 Click here to see how sin function is generated siby sebastian pgt maths 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 Click here to see how cosine function is generated siby sebastian pgt maths 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 Click here to see how tangent function is generated siby sebastian pgt maths 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 siby sebastian pgt maths 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 siby sebastian pgt maths 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 siby sebastian pgt maths For any angle ο± for which the indicated functions exist: ο1 ο£ sin ο± ο£ 1 ο1 ο£ cos ο± ο£ 1 sec ο± ο£ ο1 or sec ο± ο³ 1 tan ο± and cot ο± can take any real number csc ο± ο£ ο1 or csc ο± ο³ 1. Note that sec ο± and csc ο± are never between ο1 and 1 siby sebastian pgt maths siby sebastian pgt maths siby sebastian pgt maths 1.sin( π π − π) = ππππ π 2.cos( π − π) = sinx π 3.tan( π − π) = cotx siby sebastian pgt maths π π 4.sin( + π) = ππππ π 5.cos( π + π) = - sinx π 6.tan( π + π) = - cotx siby sebastian pgt maths 7.sin(π − π) = π¬π’π§π± 8.cos(π − π) = -cosx 9.tan(π − π) = - tanx siby sebastian pgt maths 10.sin(π + π) = −π¬π’π§π± 11.cos(π + π) = -cosx 12.tan(π + π) = tanx siby sebastian pgt maths 13.sin( ππ π − π) = −ππππ ππ 14.cos( π − π) = -sinx ππ 15.tan( π − π) = cotx siby sebastian pgt maths ππ 16. sin( + π) = −ππππ π ππ 17 .cos( + π) = sinx π ππ 18 .tan( + π) = - cotx π siby sebastian pgt maths 19.sin(ππ − π) = −π¬π’π§π± 20.cos(ππ − π) = cosx 21.tan(2π − π) =-tanx siby sebastian pgt maths 1. sin2x +cos2x =1 2. 1+tan2x =sec2x 3. 1+cot2x =cosec2x siby sebastian pgt maths 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 5.tan(x + y) = ππππ+ππππ π−ππππππππ 6.tan(x – y) = ππππ−ππππ π+ππππππππ 7.cot(x + y) = ππππππππ −π ππππ+ππππ 8.cot(x - y) = ππππππππ+π ππππ−ππππ siby sebastian pgt maths 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 SUM OR DIFFERENCE AS PRODUCT π+π π−π 1.sinx + siny = 2sin( )πππ( ) π π π+π π−π 2.sinx – siny = 2cos( )πππ( ) π π π+π π−π 3.cosx + cosy = 2cos( )πππ( ) π π π+π π−π 4.cosx – cosy = - 2sin( )πππ( ) π π siby sebastian pgt maths MULTIPLE ANGLES 1.sin2x = 2sinxcosx = πππππ π+ππππ π 2.cos2x = cos2x – sin2x = 2cos2x – 1 = 1 – 2sin2x = siby sebastian pgt maths π−ππππ π π+ππππ π 3.tan2x = πππππ π−ππππ π 4.sin3x = 3sinx – 4sin3x 5.cos3x = 4cos3x – 3cosx 6.tan3x = πππππ − ππππ π π−πππππ π siby sebastian pgt maths SUB MULTIPLE ANGLES π π 1.sinx = 2sin πππ π π ππ ππ 2.cosx = πππ − πππ π π ππ 3.1- cosx = 2πππ π ππ 4.1+cosx = 2πππ π siby sebastian pgt maths GENERAL SOLUTIONS 1.sinx =0 then x= nπ , n∈ π 2.cosx = 0 then x=(2n + π 1) , π n∈ π 3.tanx =0 then x= nπ , n∈ π siby sebastian pgt maths 4.Sinx = siny then,x = nπ + (−π)π π, n ∈ π 5.cosx =cosy then, π = πππ ± y ,n ∈ π 6.tanx = tany then x= nπ + π, n ∈ π siby sebastian pgt maths Sine Rule π π π = = ππππ¨ ππππ© ππππͺ siby sebastian pgt maths Cosine Rule cosA = ππ +ππ −ππ πππ cosB = ππ +π−ππ πππ cosC = ππ +ππ −ππ πππ siby sebastian pgt maths Practice Practice & Practice Until you get it. …….. siby sebastian pgt maths
