In this post we will try to understand how the value of trigonometry function change with the change in the position of quadrants. We have already discussed in the previous post that there are four number of quadrants in two dimensional plane. Here we will see the change in trigonometry function value with quadrant rotation.
Trigonometry function value in Quadrant rotation
Change in Value after 2π
The sign of all the trigonometry function will remain same when they are represented to the multiple of 2π
For example
where x is acute angle less than 90 degree
You can observe from above that the sign of all the trigonometry function is positive when the angle is written in the form of 2π.
Why all values are positive with 2π ?
All the trigonometry functions are positive because with 2π +x, the trigonometry function lies in first quadrant and we have already studied that all the function in first quadrant is positive.
Let us understand the concept with the help of below illustration
Consider θ = 75 degree
let us see how the trigonometric function behaves with θ = 75 degree
All the functions are positive because they lie in first quadrant.
Angles with multiples of 2nπ
sin (2nπ +x) ==> sin x
cos (2nπ +x) ==> cos x
tan (2nπ +x) ==> tan x
cosec (2nπ +x) ==> cosec x
sec (2nπ +x) ==> sec x
cot (2nπ +x) ==> cot x
where;
n is any natural number
x is angle less than 90 degree
Example
Find the value of sin (1110)
Solution
we know that 2π is a radian value
2π can be converted degree by putting value of {π = 180}
so,
2π => 2 * 180 => 360 degree
Now sin (1110) can be written as following:
Sin (1110) = Sin((3 * 360) + 30)
From the figure you can see that the value of sin function lies in first quadrant
Sin((3 * 360) + 30) = sin 30 = 1/2
Conclusion
Any trigonometry function that can be written in the multiple of 2π or 360 degree will return positive value
i.e.
tan (2π + x) = tan x
tan (360 +x) = tan x
Solved Examples
Example 01
Find the value of cosec (– 1410°)
Solution
cosec (– 1410°) => – cosec ( 1410°) { because cosec is negative in 4th quadrant}
=> – cosec ( 4* 360°- 30)
=> – cosec (– 30°) {cosec is negative in 4th quadrant}
=> – (– cosec 30°)
==> cosec 30°
==> 2
Example 02
Find the value of sin 765°
Solution
sin 765° = sin ((2 * 360°) + 45°) {angle lies in first quadrant}
==> sin ((2 * 360°) + 45°)
==> sin (45°)
==> \frac { 1 }{ \sqrt { 2 } }
Example 03
Find the value of tan(\frac { 19\pi }{ 3 } )
Solution
tan (19π/3) ==> tan (6π +π/3) [angle lies in first quadrant where all functions are positive]
==> tan (6π +π/3)
==>tan (π/3)
we know that π = 180°
==> tan (180°/3)
==> tan (60°)
==> \sqrt { 3 }
Example 04
Find the value of cot(-\frac { 15\pi }{ 4 } )
Solution
\quad cot(-\frac { 15\pi }{ 4 } )\quad =\quad -cot(\frac { 15\pi }{ 4 } )\quad \quad \quad \quad (Cot\quad is\quad negative\quad in\quad fourth\quad quadrant)
==> – cot (4π – π/4)
The angle lies in 4th quadrant where value of cot is negative such that: cot (2π – x) => – cot (x)
==> – (- cot π/4)
==> cot π/4
==> 1
Change in Value after π/2 and 3π/2 angle rotation
we have already discussed what will happen to trigonometry function when the angle is in the form of 2π, now we will discuss the value of trigonometry function when we shift the value of angle by π/2 and 3π/2
While shifting the angle by π/2 and 3π/2 remember the following rules
a. There will be change in trigonometry function
Sin Function ———–> Cos Function
Cos Function ———–> Sin Function
Tan Function ———–> Cot Function
Cot Function ———–> Tan Function
Sec Function ———–> Cosec Function
Cosec Function ———–> Sec Function
b. The sign of the function will depend on the quadrant rule which we have discussed in previous chapter
1st Quadrant —> All function will be positive
2nd Quadrant –> Sin and Cosec Function will be positive
3rd Quadrant —> Tan and Cot function will be positive
4th Quadrant —> Cos and Sec Function will be positive
The above two rules can be summarized by using following expressions
–> Expressions in the form of (π/2 – x) { Angle lies in the first quadrant }
–> Expressions in the form of (π/2 + x) {Angle lies in second quadrant}
–> Expressions in the form of (3π/2 + x) {Angle lies in fourth quadrant}
–> Expressions in the form of (3π/2 – x) {Angle lies in third quadrant}
We have covered all the possible combination of angles having π/2 and 3π/2 combination. Please do not try to memorize the concept, instead try to understand it and i will promise you will never forget the concept for a long time.
Change in values by π rotation
While shifting the angle by π remember the following rules
1. There will be no change in the trigonometry function
2. The sign of the function depends on the quadrant rule
Below are all the expressions associated with (π + x) {angle lies in third quadrant}
Below are all the expressions associated with (π – x) {angle lies in second quadrant}
