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Sine_curves_ final ppt.pptx
1. Sine curve and its
transformations
Sandeepan Narayan Behera
Class-XI,
Section- E
2. BASIC PROPERTIES OF SINE FUNCTION
Sine is a trigonometric function. It is defined as p/h in a right angle triangle, where p is the
perpendicular and h is the hypotenuse.In triangle ABC,sin A=BC/AB.
Sin A is a periodic function, with period 2 π
Sin A is an odd function. Sin (-A)= -Sin A
Domain of Sin x : ( -∞ , +∞ )
Range of Sin x : [ -1 , 1 ]
3. GRAPH OF SINX
If, sin θ = sin α
Then, General Solution of sine function :
θ = nπ + (-1)
n
α, where α ∈ [-π/2, π/2]
4. The values of sin A for particular values
of angle A
A (degree) Sin A A (degree) Sin A
0 0 180 0
30 1/2 210 -1/2
45 1/√2 225 -1/√2
60 √3/2 240 -√3/2
90 1 270 -1
120 √3/2 300 -√3/2
135 1/√2 315 -1/√2
150 ½ 330 -½
180 0 360 0
5. HORIZONTAL PHASE SHIFT OF SIN CURVES
The constant c in the general equations y = sin(x – c) creates horizontal shifts of
the basic sine curves.If y=sin(x-c),it shifts c units right and if y=sin(x+c),then the
graph shifts –c units or c units towards left.
In the given figure,the red part of the graph represents the
actual sin curve And the blue part represents the horizontally
shifted sin curve sin (x-π/4).It has shifted /4 units right.
6. ALLIED ANGLES
α ( in degrees) Sin α
90–θ +Cosθ
90+θ +Cosθ
180–θ +Sinθ
180+θ –Sinθ
270–θ -Cosθ
270+θ –Cosθ
360–θ –Sinθ
General conversions
1.sin(n180±A)= ±sin A(+ve for 1,2 Quadrant in Cartesian
plane and –ve for 3,4 Quadrant in Cartesian plane).
2.Sin(n90±A)=±cos A(+ve for 1,2 quadrant in Cartesian
plane and –ve for 3,4 quadrant in Cartesian plane)
7. EXAMPLES BASED ON THE ABOVE
CONCEPTS
Example 1- Find the value of sin 225°
Solution-
sin(180+45)=-sin 45 °(Since , if something is added to or subtracted to 180 °,then sin function doesn’t
change only sign of sine function changes).
-Sin 45= -1/√2
Example 2- Find the value of sin 150°
Solution-
Sin(90 + 60) = +cos 60° ( Since 150° lies in the second quadrant)
Sin 150° = cos 60° = 1/2
8. Vertical shift of sine curve
If , y= a + sin x, then the curve shifts upwards by a unit and if , y= a- sinx , then curve shifts
downwards by –a units
Other forms are y= - a + sinx , y= -a - sinx
9. Sine curve with f(x)= k sin x transformation
Y= a sin x form changes the amplitude of sin x to from 1 to k.The
range of ksinx is [-k,k].The crest and trough changes as k times -1
or 1
13. APPLICATIONS
Astronomy :
It helps in determining the distance between the stars and planets. The tables
help in locating the position of a sphere and this kind of trigonometry is called
spherical trigonometry.
Construction :
Trigonometry is used in construction sites in measuring ground surfaces and
fields. When constructing a building, aspects like roof inclination, rook slopes,
perpendicular and parallel walls, light angles, sun shading, etc, require
trigonometry.
Satellite Navigation System :
The satellite navigation system also uses trigonometric functions such as the
law of cosine to calculate simple equations to help in locating the satellites in
the earth's orbit 24 hours.