In class 10,CBSE,some basic concepts and ideas about trigonometry was introduced [Angle greater than zero and less than 90].
Here,we shall try to compute the value of trigonometric functions whose angle is greater than 90.
Angles can be measured in different units.One such unit that we are familiar with is the unit "Degree".There are two other important Units-Radian and Gradian.
*When a circle is divided into 360 equal parts,each part represents 1degree.
*In a circle,If the length of the arc is equal to the radius of the circle,then the angle subtended is 1 radian
{These are the units of plane angle.There are also units of solid angle(steradian)}
NOTE: 2π radians = 360 degrees.
An other important relation is:
θ=L/r
'θ' is the angle in radians,'L' is the length of the arc of the circle,'r' is the radius of the circle.
Now let us try to learn some rules that will enable us to compute the value of some trigonometric functions
Let us first work with the degree measure.
In class 10, we have learnt the values of standard angles.
Any angle can can be written in the form (a+b),'a' being the integral multiples of 90 degree.
For example,
120=(90+30) or (180-60)
150=(180-30)
If the angle that is to be written is an odd multiple of 90,you change the function. {Odd angles include 90,270,450.....}
The function changes in the following manner:
sin----cos
cos---sin
tan---cot
cot---tan
sec---cosec
cosec---sec
For example,sin(90+θ) is written as cosθ.
If the angle is even multiples of 90(180,360 etc),the function is retained as it is.
For example, tan(180+θ)=tanθ.
The same thing holds good for radian measure.
For example, tan(π+π/3)=tan(π/3).
There is another rule which has to be followed while computing the values of the trigonometric functions-The "ASTC" rule.
In the first quadrant,all the trigonometric functions are positive.
In the second quadrant,only sin and cosecant are positive and the rest are negative.
In the third quadrant,only tan and cot are positive and the rest are negative.
In the forth quadrant,only cos and sec are positive,rest are negative.
The two rules are very important.
Examples:
1) cos(90+θ)= -sinθ
since it is an odd multiple of 90 , cos changes to sin. {90+θ} is an angle that lies in the second quadrant,and in the second quadrant,the original function cos is negative.
2) sin(180+θ)= -sinθ.
3) cot(180+θ)=cotθ
using the rules,the following are possible:
sin(-θ)= -sinθ(sin is negative in the 4th quadrant.)
cos(-θ)= cosθ (cos is + in the 4th quadrant)
tan(-θ)=-tanθ
cosec(-θ)= -cosecθ
sec(-θ)= secθ
cot(-θ)= -cotθ
These are the basics for computation of the trigonometric functions.There are so many identities and formulas and equations in trigonometry. It is important to know this before you learn the identities.
Trigonometry is awesome if you understand...
Peace out...!!
Here,we shall try to compute the value of trigonometric functions whose angle is greater than 90.
Angles can be measured in different units.One such unit that we are familiar with is the unit "Degree".There are two other important Units-Radian and Gradian.
*When a circle is divided into 360 equal parts,each part represents 1degree.
*In a circle,If the length of the arc is equal to the radius of the circle,then the angle subtended is 1 radian
{These are the units of plane angle.There are also units of solid angle(steradian)}
NOTE: 2π radians = 360 degrees.
An other important relation is:
θ=L/r
'θ' is the angle in radians,'L' is the length of the arc of the circle,'r' is the radius of the circle.
Now let us try to learn some rules that will enable us to compute the value of some trigonometric functions
Let us first work with the degree measure.
In class 10, we have learnt the values of standard angles.
Any angle can can be written in the form (a+b),'a' being the integral multiples of 90 degree.
For example,
120=(90+30) or (180-60)
150=(180-30)
If the angle that is to be written is an odd multiple of 90,you change the function. {Odd angles include 90,270,450.....}
The function changes in the following manner:
sin----cos
cos---sin
tan---cot
cot---tan
sec---cosec
cosec---sec
For example,sin(90+θ) is written as cosθ.
If the angle is even multiples of 90(180,360 etc),the function is retained as it is.
For example, tan(180+θ)=tanθ.
The same thing holds good for radian measure.
For example, tan(π+π/3)=tan(π/3).
There is another rule which has to be followed while computing the values of the trigonometric functions-The "ASTC" rule.
In the second quadrant,only sin and cosecant are positive and the rest are negative.
In the third quadrant,only tan and cot are positive and the rest are negative.
In the forth quadrant,only cos and sec are positive,rest are negative.
The two rules are very important.
Examples:
1) cos(90+θ)= -sinθ
since it is an odd multiple of 90 , cos changes to sin. {90+θ} is an angle that lies in the second quadrant,and in the second quadrant,the original function cos is negative.
2) sin(180+θ)= -sinθ.
3) cot(180+θ)=cotθ
using the rules,the following are possible:
sin(-θ)= -sinθ(sin is negative in the 4th quadrant.)
cos(-θ)= cosθ (cos is + in the 4th quadrant)
tan(-θ)=-tanθ
cosec(-θ)= -cosecθ
sec(-θ)= secθ
cot(-θ)= -cotθ
These are the basics for computation of the trigonometric functions.There are so many identities and formulas and equations in trigonometry. It is important to know this before you learn the identities.
Trigonometry is awesome if you understand...
Peace out...!!
