9th Math Chapter6 Trigonometry Notes
9th Math Chapter6 Trigonometry Notes for Federal board as well as Punjab boards New book. We will discuss about types of triangles, how to measure an angle, angle in standard position, system of measurements of an angle and their examples also.
Tri means three, gone means angle and metry means measurement. Trigonometry means study of plane figures having three angles.
<———————————–> Line
(There is no end point in a line. So it is not measureable)
————————————– Line segment
(Line segment has two end points and line segment is a part of line. Hence it is measurable).
————————————-> Ray
(Ray has only one end point. Like line its also not measurable)
Definition of angle:- Two rays starting from a common point form an angle.
Trigonometry is a branch of geometry is which we study triangles.
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Types of triangle w.r.t sides.
There are three types of triangles with respect to sides.
(i) :- Equilateral triangle.
A triangle is which all the sides are equal in length. For example 5cm, 5cm and 5cm.
(ii) :- Isosceles triangle
A triangle is which two sides are equal in length. For example 5cm, 5cm and 4cm.
(iii) :- scalene triangle
A triangle is which all the sides are not equal in length. For example 5cm, 4cm and 2cm.
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Types of triangle w.r.t angles.
There are three types of triangles with respect to angles.
(i) :- Right angle triangle
A triangle is which there is as angle of $$90^\circ$$.
(ii) :- Obtuse angle triangle
A triangle is which there is an angle of greater than $$90^\circ$$.
(iii) :- Acute angle triangle
A triangle is which all the angles are acute i.e all the angles are less than $$90^\circ$$.
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NOTE:-
(i):- In any triangle, two angles must be acute i.e less than 90 degrees.
(ii):- Sum of all the angles in a triangle will be 180 degrees.
Measurement of an Angle.
Union of two rays having common end point form an angle.
An angle is positive if it is measured is anti-clockwise direction and it will be negative if it is measured is clockwise direction.
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Angle is standard position.
An angle when drawn is in standard position in the cartesian coordinate system if its vertex is at the origin and its initial ray is directed along positive x-axis.
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System of measurement of angle.
There are three types of systems of measurement of an angle.
(i) :- Sexagesimal system ( English or DMS system ).
(ii) :- Radian system ( circular system ).
(iii) :- French system ( centesimal system ).
In our syllabus only first two systems are included.
Lets check sexagesimal system first.
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Definition of Circumference.
Boundary of a circle is called its circumference.
Definition of Arc.
A part of circumference of a circle is called an arc.
Divide the circumference of a circle into 360 equal parts. Each part is making a central angle of one degree denoted by $$1^\circ$$. Divide 1 degree into 60 equal parts. Each part is called minute denoted by $$1’$$. $$1^\circ=\;60’$$. Divide 1 minute into 60 equal parts. Each part is called second denoted by $$1”$$.$$1’\;=\;60”$$.
$$1^\circ=3600”$$
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Lets check radian system now.
Definition of one radian:-
A central angle by an arc, whose length is equal to the radius of the circle.
If l = r, then q = 1 rad
Radian :-
Ratio between length of arc and the radius of the same circle.
$$180^\circ\;=\;\mathrm\pi\;\mathrm{rad}\\\mathrm{or}\;\;1^\circ\;=\;\frac{\mathrm\pi}{180}\;\mathrm{rad}$$
$$After\;putting\;the\\value\;of\;\mathrm\pi,\;\mathrm{we}\;\mathrm{get}\\\\1^\circ\;=\;0.017\;rad$$
$$How\;to\;convert\;30^\circ\\into\;radians$$
Formula:- $$1^\circ=\frac{\mathrm\pi}{180}\;rad$$
Multiply both the side by 30.
$$30^\circ\;=\;30(\frac{\mathrm\pi}{180})\;rad\\30^\circ\;=\;\frac{\mathrm\pi}6\;rad$$
Similarly
$$\mathrm\pi\;\mathrm{rad}\;=\;180^\circ\\1\;\mathrm{rad}\;=\;\frac{180^\circ}{\mathrm\pi}$$
$$\mathrm{After}\;\mathrm{puting}\;\mathrm{the}\;\mathrm{value}\\\mathrm{of}\;\mathrm\pi,\;\mathrm{we}\;\mathrm{get}\\\\1\;\mathrm{rad}\;=\;57^\circ\;17’\;45”$$
$$1\;\mathrm{rad}\;=\;57.2958^\circ$$
$$How\;to\;convert\;\frac{\mathrm\pi}6rad\\into\;degrees$$
Formula:-
$$1\;rad\;=\;\frac{180^\circ}{\mathrm\pi}$$
$$\frac{\mathrm\pi}6\;rad\;=\frac{\mathrm\pi}6(\;\frac{180^\circ}{\mathrm\pi})\\\\\frac{\mathrm\pi}6\;rad\;=30^\circ$$
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