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}