Trigonometry And Differentiation

Trigonometry And Differentiation

Trigonometric Ratios and Differentiation
 Trigonometry:Trigonometry plays an important role in physics, which takes you to a different level in understanding physics
 Introduction: 
The word “trigonometry” is a Greek word. Its mean “measurement of a triangle”. Therefore trigonometry is that branch of mathematics concerned with the measurement of sides and angle of a plane triangle and the investigations of the various relations which exist among them. Today the subject of trigonometry also includes another distinct branch which concerns itself with properties relations between and behavior of trigonometric functions. 
The importance of trigonometry will be immediately realized when its applications in solving problem of mensuration, mechanics physics, surveying and astronomy are encountered. 
Types of Trigonometry: 
There are two types of trigonometry 
1. Plane Trigonometry : Plane trigonometry is concerned with angles, triangles and other figures which lie in a plane. 
2. Spherical Trigonometry:Spherical Trigonometry is concerned with the spherical triangles, that is, triangles lies on a sphere and sides of which are circular arcs. 
Angle: 
An angle is defined as the union of two non-collinear rays which have a common start or end-points. 
An angle is also defined as it measures the rotation of a line from one position to another about a fixed point on it.

position OP is called terminal line or generating line(position) of \( \angle XOP \).
If the terminal side resolves in anticlockwise direction the angle described is positive as shown in figure below left. If terminal side resolves in clockwise direction, the angle described is negative as shown in figure right

  
The fundamental unit of angle measure is the degree of arc. By definition, when a circle is divided into 360 equal parts, then 
    One degree = 1/ 360 th part of a circle.
    Therefore, one full circle = 360 degrees. 
    The symbol of degrees is denoted by ( )⁰ . 
The degree is further subdivided in two ways, 
    1 degree = 60 minutes=60’ 
    1 minute = 60 seconds=60’’
Relation between Degree and Radian Measure:
Consider a circle of radius r, then the circumference of the circle is .
By definition of radian, 
An arc of length \( 2\pi r \) subtends an angle =\( 2\pi \) radian
Also an arc of length \( 2\pi r \) subtends an angle = 360^o
    Then \( 2\pi \) radians = 360^o
    Or \( \pi \) radians = 180^o
Trigonometric Function and Ratios:
In the figure OMP is a right angled triangle(Let the initial line OX revolves and trace out an angle \( \theta \)¸ Take a point P on the final line. Draw perpendicular PM from P on OX: \( \angle XOP = \theta \)), We can form the six ratios as follows

\( \sin \theta = \frac{a} {c} = \frac{{MP}} {{OP}} = \frac{{Perpendicular}} {{Hypotenuse}} \)
\( \cos \theta = \frac{b} {c} = \frac{{OM}} {{OP}} = \frac{{Base}} {{Hypotenuse}} \)
\( \tan \theta = \frac{a} {b} = \frac{{MP}} {{OM}} = \frac{{Perpendicular}} {{Base}} = \frac{{\sin \theta }} {{\cos \theta }} \)
\( \cot \theta = \frac{b} {a} = \frac{{OM}} {{MP}} = \frac{{Base}} {{Perpendicular}} = \frac{1} {{\tan \theta }} = \frac{{\cos \theta }} {{\sin \theta }} \)
\( \sec \theta = \frac{c} {b} = \frac{{OP}} {{OM}} = \frac{{Hypotenuse}} {{Base}} = \frac{1} {{\cos \theta }} \)
\( \cos ec = \frac{c} {a} = \frac{{OP}} {{MP}} = \frac{{Hypotenuse}} {{Perpendicular}} = \frac{1} {{\sin \theta }} \)
Note: For the given angle Perpendicular can be called as opposite side and Base can be called as adjacent side

Trigonometric Ratios of Particular Angles: 
    In the above picture if triangle is drawn for an angle \( \theta = 30^0 \) for which ever side lenghts we get always Hypotenuse length is twice to perpendicular length. It means \( \frac{{Perpendicular}} {{Hypotenuse}} = \frac{{MP}} {{OP}} = \frac{a} {c} = \frac{1} {2} = \sin 30^0 = \text{constant} \)

It says if a=1 unit, then c=2 units. By using Pythagorean theorem 
    c²=a²+b², we get b=\( \sqrt 3 \)
 Therefore 

\( \cos 30^0 = \frac{b} {c} = \frac{{\sqrt 3 }} {2},\tan 30^0 = \frac{a} {b} = \frac{1} {{\sqrt 3 }},\cot 30^0 = \frac{b} {a} = \sqrt 3 ,\sec 30^0 = \frac{c} {b} = \frac{2} {{\sqrt 3 }},\cos ec30^0 = \frac{c} {a} = 2 \)
similarly if we verify these ratios always constant for different angles, independent of side length of right angled triangle.


Quadrants: 
    Two mutually perpendiculars straight lines XOX’ and YOY’ divide the plane into four equal parts, each part is called quadrant. Thus XOY, YOX’, X’OY’ and Y’OX are called the Ist, IInd, IIIrd and IVth quadrants respectively.

In anti-clockwise direction in first quadrant the angle vary from 0^o to 90^o, in second quadrant from 90^o to 180^o, in third quadrant vary from 180^o to 270^o and in fourth quadrant the angle vary from 270^o to 360^o. 
Rectangular Co-ordinates and Sign Convention: 
    In plane geometry the position of a point can be fixed by measuring its perpendicular distance from each of two perpendicular called co-ordinate axes. The horizontal line (x-axis) and the vertical line(y-axis). 
        Distance measured from the point O in the direction OX and OY are regarded as positive, while in the direction of OX’ and OY’ are considered negative. 
Signs of Trigonometric Functions: 
    The trigonometric ratios discussed above have different signs in different quadrants. We can remember the sign of trigonometric function by “ASTC” (funnily All Silver Tea Cups) Rule. In “ASTC” A stands for All and C stands for cos (cosine) and T stands for tan(Tangent) and S stands for Sine. i.e.
1)  In first quadrant sign of all the trigonometric functions are positive i.e., sin, cos, tan, Cot, Sec, Cosec all are positive. 
2)  In second quadrant sine and its inverse cosec are positive. The remaining four trigonometric function i.e., cos, tan, cot, sec are negative. 
3)  In third quadrant tan and its reciprocal cot are positive the remaining four function i.e., Sin, cos, sec and cosec are negative. 
4)  In fourth quadrant cos and its reciprocal sec are positive, the remaining four functions i.e., sin, tan, cot and cosec are negative.

Finding the values of Trigonometric ratios when \( \mathbf{\theta > 90}^\mathbf{0} \):
Every angle A can be reduced to the form ‘\( A = n \times 90^0 \pm \theta \)’ where \( 0^0 \leqslant \theta < 90^0 \).
(i) If n is a odd integer, we have
\( \sin \left( {n \times 90^0 \pm \theta } \right) = \, \pm \cos \,\theta ,\,\,\cos \,\left( {n \times 90^0 \pm \theta } \right) = \, \pm \,\sin \,\theta ,\,\,\tan \,\left( {n \times 90^0 \pm \theta } \right) = \pm \,\cot \,\theta \)
\( \cot \,\left( {n \times 90^0 \pm \theta } \right) = \pm \,\tan \theta ,\,\,\,\cos ec\left( {n \times 90^0 \pm \theta } \right) = \, \pm \sec \,\theta ,\,\,\sec \,\left( {n \times 90^0 \pm \theta } \right) = \, \pm \,\cos ec\,\theta ,\,\, \)
i.e., sin changes to cos, cos changes to sin, tan changes to cot, cot changes to tan, sec changes to cosec and cosec changes to sec. 
(ii) If n is an even integer, we have    
\( \sin \left( {n \times 90^0 \pm \theta } \right) = \, \pm \sin \,\theta ,\,\,\cos \,\left( {n \times 90^0 \pm \theta } \right) = \, \pm \cos \,\theta ,\,\,\tan \,\left( {n \times 90^0 \pm \theta } \right) = \pm \,\tan \,\theta \),
\( \cos ec\left( {n \times 90^0 \pm \theta } \right) = \, \pm \cos ec\,\theta ,\,\,\sec \,\left( {n \times 90^0 \pm \theta } \right) = \, \pm \,\sec \,\theta ,\,\,\cot \,\left( {n \times 90^0 \pm \theta } \right) = \pm \,\cot \,\theta \)

i.e., sin remains sin, cos remains cos, tan remains tan, cosec remains cosec, sec remains sec and cot remains cot. 
(iii) Use original (given) ratio to find + or – sign in the R.H.S. of the equations in (i) and (ii) making use of the phrase All Silver Tea Cups.
Trigonometrical ratios of negative angles :
sin(-\( \theta \))=-sin\( \theta \),          cos(-\( \theta \))=cos\( \theta \),          tan(-\( \theta \))=-tan\( \theta \) 
cosec(-\(\theta \))=-cosec\(\theta \),    sec(-\(\theta \))=sec\(\theta \),        cot(-\(\theta \))=-cot\(\theta \) 
Trigonometrical ratios of complementary angles :
sin (90 -\(\theta \)) = cos\(\theta \);    cos (90 -\(\theta \)) = sin\(\theta \);    tan (90 -\(\theta \)) = cot\(\theta \).
Trigonometrical ratios of supplementary angles :
sin (180 -\(\theta \)) = sin\(\theta \),    cos (180 -\(\theta \)) = -cos\(\theta \),    tan (180 -\(\theta \)) = - tan\(\theta \)

Important trigonometric functional relationships

Example 1: sin 120° = sin (1\( \times \)90°+30°). Here n = 1, an odd integer.
    \( \Rightarrow \) Sin changes to Cos
\( \therefore \)  120° lies in the second quadrant \( \Rightarrow \) Original (given) ratio sin is + ve.
\( \therefore \) sin 120° = sin (90°+30°) = +cos 30° =\( \sqrt 3 /2 \)
Example 2: cos 240° = cos (180° + 60°) = cos (2\( \times \)90° + 60°)
    Here  n = 2, an even integer\( \Rightarrow \) cos remains cos. 
    240° lies in the third quadrant \( \Rightarrow \) original (given) ratio cos is –ve. 
    \( \therefore \)cos 240° = – cos 60° = –1/2. 
Example 3: tan (–300°) = – tan 300° = –tan (390° + 30°). 
    Here n = 3, an odd integer. tan 300° lies in the fourth quadrant 
    \( \Rightarrow \) original (given) ratio tan is –ve
    \( \therefore \) tan (–300°) = –(–cot30°) =\( \sqrt 3 \).

The most important values of trigonometric functions.


Important Trigonometrical formulae:
Pythagorean Identities
1)    sin²\( \theta \) + cos²\( \theta \) = 1 
2)    1+ tan²\( \theta \) = sec²\( \theta \)
3)    1+ cot²\( \theta \) = cosec²\( \theta \) 
Addition formulae :
1) \( sin\left( {A + B} \right) = \sin A\cos B + \cos A\sin B \)   
2) \( \cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B \)   
3) \( \tan \left( {A + B} \right) = \frac{{\tan A + \tan B}} {{1 - \tan A\tan B}} \) 

Subtraction formulae :
1) \( sin\left( {A - B} \right) = \sin A\cos B - \cos A\sin B \)   
2) \( \cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B \)   
3) \( \tan \left( {A - B} \right) = \frac{{\tan A - \tan B}} {{1 + \tan A\tan B}} \)   
Sum and difference formulae :
1) \( \sin A + \sin B = 2\sin \left( {\frac{{A + B}} {2}} \right)\cos \left( {\frac{{A - B}} {2}} \right) \)   
2) \( \sin A - \sin B = 2\cos \left( {\frac{{A + B}} {2}} \right)\sin \left( {\frac{{A - B}} {2}} \right) \)   
3) \( \cos A + \cos B = 2\cos \left( {\frac{{A + B}} {2}} \right)\cos \left( {\frac{{A - B}} {2}} \right) \)   
4) \( \cos A - \cos B = - 2\sin \left( {\frac{{A + B}} {2}} \right)\sin \left( {\frac{{A - B}} {2}} \right) \) \( = 2\sin \left( {\frac{{A + B}} {2}} \right)\sin \left( {\frac{{B - A}} {2}} \right) \)  
Formulae related to multiple angles :
1)    sin2A = 2sinAcosA; sinA = 2sin\( \tfrac{A} {2} \) cos\( \tfrac{A} {2} \)
2)    cos2A = cos² A – sin² A = 2cos²A –1=1–2sin² A
3)    cosA =2cos²\( \tfrac{A} {2} \)–1=1–2sin²\( \tfrac{A} {2} \) ; 1+cosA = 2cos² \( \tfrac{A} {2} \)
4)    1–cosA = 2sin²\( \tfrac{A} {2} \)
5)    \( \tan 2A = \frac{{2\tan A}} {{1 - \tan ^2 A}} \)
Note: When the angle is small, sin \( \theta \)=\( \theta \) when \( \theta \) is expressed in radians
    cos\( \theta \) =\( 1 - \frac{{\theta ^2 }} {2} \) and tan\(\theta\) =\(\theta\) 
    sin1⁰= tan1⁰=\( \frac{\pi } {{180}} \) becausep \( \pi \) radians =180⁰