sin (π/2 – x) = cos x

cos (π/2 – x) = sin x

sin (π/2 + x) = cos x

cos (π/2 + x) = – sin x

sin (3π/2 – x)  = – cos x

cos (3π/2 – x)  = – sin x

sin (3π/2 + x) = – cos x

cos (3π/2 + x) = sin x

sin (π – x) = sin x

cos (π – x) = – cos x

sin (π + x) = – sin x

cos (π + x) = – cos x

sin (2π – x) = – sin x

cos (2π – x) = cos x

sin (2π + x) = sin x

cos (2π + x) = cos x

Let us learn how to find and remember these values

We follow two rules

1. If the angle is multiple of π/2, i.e. π/2, 3π/2, 5π/2,

then 

sin becomes cos

cos becomes sin

If the angle is multiple of π, i.e. π, 2π, 3π,

then 

sin remains sin

cos remains sin

 

2.The sign depends on the quadrant angle is in.

sin ( π /2 – x)

 

Since it is π/2,

sin will become cos

 

Here x is an acute angle

So, π/2 – x = 90 – x is an angle in 1st quadrant.

sin (pi 2  x).jpg

Since sin is positive in 1st quadrant

So, sign will be positive

 

∴ sin (π/2 – x) = cos x

 

cos ( π /2 – x)

 

Since it is π/2,

cos will become sin

 

Here x is an acute angle

So, π/2 – x = 90 – x is an angle in 1st quadrant.

cos (pi 2  x).jpg

Since cos is positive in 1st quadrant

So, sign will be positive

 

∴ cos (π/2 – x) = sin x

 

sin ( π /2 + x)

 

Since it is π/2,

sin will become cos

 

Here x is an acute angle

So, π/2 + x = 90 + x

90 + x is an angle which is greater than 90°, less than 180°

So, it will be in 2nd quadrant.

sin will become cos.jpg

Since sin is positive in 2nd quadrant

So, sign will be positive

∴ sin (π/2 + x) = cos x

 

cos ( π /2 + x)

 

Since it is π/2,

cos will become sin

 

Here x is an acute angle

So, π/2 + x = 90 + x

90 + x is an angle which is greater than 90°, less than 180°

So, it will be in 2nd quadrant.

cos will become sin.jpg

Since cos is negative in 2nd quadrant

So, sign will be negative

∴ cos (π/2 + x) = – sin x

 

sin (3 π /2 – x)

 

Since it is 3π/2,

sin will become cos

 

Here x is an acute angle

So, 3π/2 – x = 270 – x

270 – x is an angle which is greater than 180°, less than 270°

So, it will be in 3rd quadrant.

sin (3pi 2  x).jpg

Since sin is negative in 3rd quadrant

So, sign will be negative

∴ sin (3π/2 – x) = – cos x

 

cos (3 π /2 – x)

 

Since it is 3π/2,

cos will become sin

 

Here x is an acute angle

So, 3π/2 – x = 270 – x

270 – x is an angle which is greater than 180°, less than 270°

So, it will be in 3rd quadrant.

cos (3pi 2 x).jpg

Since cos is negative in 3rd quadrant

So, sign will be negative

∴ cos (3π/2 – x) = – sin x

 

sin (3 π /2 + x)

 

Since it is 3π/2,

sin will become cos

 

Here x is an acute angle

So, 3π/2 + x = 270 + x

270 + x is an angle which is greater than 270°, less than 360°

So, it will be in 4th quadrant.

sin (3pi plus x).jpg

Since sin is negative in 4th quadrant

So, sign will be negative

∴ sin (3π/2 + x) = – cos x

 

cos (3 π /2 + x)

 

Since it is 3π/2,

cos will become sin

 

Here x is an acute angle

So, 3π/2 + x = 270 + x

270 + x is an angle which is greater than 270°, less than 360°

So, it will be in 4th quadrant.

cos (3pi plus x).jpg

Since cos is positive in 4th quadrant

So, sign will be positive

∴ cos (3π/2 + x) = sin x

 

sin ( π x)

 

Since it is π,

sin will remain sin

 

Here x is an acute angle

So, π – x = 180 – x

180 – x is an angle which is greater than 90°, less than 180°

So, it will be in 2nd quadrant.

sin (pi minus x).jpg

Since sin is positive in 2nd quadrant

So, sign will be positive

∴ sin (π – x) = sin x

 

cos ( π x)

 

Since it is π,

cos will remain cos

 

Here x is an acute angle

So, π – x = 180 – x

180 – x is an angle which is greater than 90°, less than 180°

So, it will be in 2nd quadrant.

cos (pi minus x).jpg

Since cos is negative in 2nd quadrant

So, sign will be negative

∴ cos (π – x) = – cos x

 

sin ( π + x)

 

Since it is π,

sin will remain sin

 

Here x is an acute angle

So, π + x = 180 + x

180 + x is an angle which is greater than 180°, less than 270°

So, it will be in  3rd quadrant.

sin (pi plus x).jpg

Since sin is negative in 3rd quadrant

So, sign will be negative

∴ sin (π + x) = – sin x

 

cos ( π + x)

 

Since it is π,

cos will remain cos

 

Here x is an acute angle

So, π + x = 180 + x

180 + x is an angle which is greater than 180°, less than 270°

So, it will be in  3rd quadrant.

cos (pi plus x).jpg

Since cos is negative in 3rd quadrant

So, sign will be negative

∴ cos (π + x) = – cos x

 

sin (2 π x)

 

Since it is 2π,

sin will remain sin

 

Here x is an acute angle

So, 2π – x = 360 – x

360 – x is an angle which is greater than 270°, less than 360°

So, it will be in  4th quadrant.

sin (2pi minus x).jpg

Since sin is negative in 4th quadrant

So, sign will be negative

 

∴ sin (2π – x) = – sin x

 

We can also prove it by

We know that sin repeats after 2π

⇒ sin (2π – x) = sin (–x)

Since sin (–x) = – sin x

∴ sin (2π – x) = – sin (x)

 

cos (2 π x)

 

Since it is 2π,

cos will remain cos

 

Here x is an acute angle

So, 2π – x = 360 – x

360 – x is an angle which is greater than 270°, less than 360°

So, it will be in  4th quadrant.

cos (2pi minus x).jpg

Since cos is positive in 4th quadrant

So, sign will be positive

 

∴ cos (2π – x) = cos x

 

We can also prove it by

We know that cos repeats after 2π

⇒ cos (2π – x) = cos (–x)

Since cos (–x) = cos x

∴ cos (2π – x) = cos x

 

sin (2 π + x)

 

Since it is 2π,

sin will remain sin

 

Here x is an acute angle

So, 2π + x = 360 + x

360 – x is an angle which is greater than 360°, less than 360° + 90° 

So, it will be in  1st quadrant.

sin (2pi plus x).jpg

Since sin is positive in 4th quadrant

So, sign will be positive

 

∴ sin (2π + x) = sin x

We can also prove it by

We know that sin repeats after 2π

⇒ sin (2π + x) = sin (x)

 

cos (2 π + x)

 

Since it is 2π,

cos will remain cos

 

Here x is an acute angle

So, 2π + x = 360 + x

360 – x is an angle which is greater than 360°, less than 360° + 90° 

So, it will be in  1st quadrant.

cos (2pi plus x).jpg

Since cos is positive in 4th quadrant

So, sign will be positive

 

∴ cos (2π + x) = cos x

We can also prove it by

We know that cos repeats after 2π

⇒ cos (2π + x) = cos (x)

  1. Chapter 3 Class 11 Trigonometric Functions
  2. Concept wise
Ask Download

Transcript

sin (π/2 – x) = cos x cos (π/2 – x) = sin x sin (π/2 + x) = cos x cos (π/2 + x) = – sin x sin (3π/2 – x)  = – cos x cos (3π/2 – x)  = – sin x sin (3π/2 + x) = – cos x cos (3π/2 + x) = sin x sin (π – x) = sin x cos (π – x) = – cos x sin (π + x) = – sin x cos (π + x) = – cos x sin (2π – x) = – sin x cos (2π – x) = cos x sin (2π + x) = sin x cos (2π + x) = cos x Let us learn how to find and remember these values We follow two rules 1. If the angle is multiple of π/2, i.e. π/2, 3π/2, 5π/2, then  sin becomes cos cos becomes sin If the angle is multiple of π, i.e. π, 2π, 3π, then  sin remains sin cos remains sin 2.The sign depends on the quadrant angle is in. sin (π/2 – x) Since it is π/2, sin will become cos Here x is an acute angle So, π/2 – x = 90 – x is an angle in 1st quadrant. Since sin is positive in 1st quadrant So, sign will be positive ∴ sin (π/2 – x) = cos x cos (π/2 – x) Since it is π/2, cos will become sin Here x is an acute angle So, π/2 – x = 90 – x is an angle in 1st quadrant. Since cos is positive in 1st quadrant So, sign will be positive ∴ cos (π/2 – x) = sin x sin (π/2 + x) Since it is π/2, sin will become cos Here x is an acute angle So, π/2 + x = 90 + x 90 + x is an angle which is greater than 90°, less than 180° So, it will be in 2nd quadrant. Since sin is positive in 2nd quadrant So, sign will be positive ∴ sin (π/2 + x) = cos x cos (π/2 + x) Since it is π/2, cos will become sin Here x is an acute angle So, π/2 + x = 90 + x 90 + x is an angle which is greater than 90°, less than 180° So, it will be in 2nd quadrant. Since cos is negative in 2nd quadrant So, sign will be negative ∴ cos (π/2 + x) = – sin x sin (3π/2 – x) Since it is 3π/2, sin will become cos Here x is an acute angle So, 3π/2 – x = 270 – x 270 – x is an angle which is greater than 180°, less than 270° So, it will be in 3rd quadrant. Since sin is negative in 3rd quadrant So, sign will be negative ∴ sin (3π/2 – x) = – cos x cos (3π/2 – x) Since it is 3π/2, cos will become sin Here x is an acute angle So, 3π/2 – x = 270 – x 270 – x is an angle which is greater than 180°, less than 270° So, it will be in 3rd quadrant. Since cos is negative in 3rd quadrant So, sign will be negative ∴ cos (3π/2 – x) = – sin x sin (3π/2 + x) Since it is 3π/2, sin will become cos Here x is an acute angle So, 3π/2 + x = 270 + x 270 + x is an angle which is greater than 270°, less than 360° So, it will be in 4th quadrant. Since sin is negative in 4th quadrant So, sign will be negative ∴ sin (3π/2 + x) = – cos x cos (3π/2 + x) Since it is 3π/2, cos will become sin Here x is an acute angle So, 3π/2 + x = 270 + x 270 + x is an angle which is greater than 270°, less than 360° So, it will be in 4th quadrant. Since cos is positive in 4th quadrant So, sign will be positive ∴ cos (3π/2 + x) = sin x sin (π – x) Since it is π, sin will remain sin Here x is an acute angle So, π – x = 180 – x 180 – x is an angle which is greater than 90°, less than 180° So, it will be in 2nd quadrant. Since sin is positive in 2nd quadrant So, sign will be positive ∴ sin (π – x) = sin x cos (π – x) Since it is π, cos will remain cos Here x is an acute angle So, π – x = 180 – x 180 – x is an angle which is greater than 90°, less than 180° So, it will be in 2nd quadrant. Since cos is negative in 2nd quadrant So, sign will be negative ∴ cos (π – x) = – cos x sin (π + x) Since it is π, sin will remain sin Here x is an acute angle So, π + x = 180 + x 180 + x is an angle which is greater than 180°, less than 270° So, it will be in  3rd quadrant. Since sin is negative in 3rd quadrant So, sign will be negative ∴ sin (π + x) = – sin x cos (π + x) Since it is π, cos will remain cos Here x is an acute angle So, π + x = 180 + x 180 + x is an angle which is greater than 180°, less than 270° So, it will be in  3rd quadrant. Since cos is negative in 3rd quadrant So, sign will be negative ∴ cos (π + x) = – cos x sin (2π – x) Since it is 2π, sin will remain sin Here x is an acute angle So, 2π – x = 360 – x 360 – x is an angle which is greater than 270°, less than 360° So, it will be in  4th quadrant. Since sin is negative in 4th quadrant So, sign will be negative ∴ sin (2π – x) = – sin x We can also prove it by We know that sin repeats after 2π ⇒ sin (2π – x) = sin (–x) Since sin (–x) = – sin x ∴ sin (2π – x) = – sin (x) cos (2π – x) Since it is 2π, cos will remain cos Here x is an acute angle So, 2π – x = 360 – x 360 – x is an angle which is greater than 270°, less than 360° So, it will be in  4th quadrant. Since cos is positive in 4th quadrant So, sign will be positive ∴ cos (2π – x) = cos x We can also prove it by We know that cos repeats after 2π ⇒ cos (2π – x) = cos (–x) Since cos (–x) = cos x ∴ cos (2π – x) = cos x sin (2π + x) Since it is 2π, sin will remain sin Here x is an acute angle So, 2π + x = 360 + x 360 – x is an angle which is greater than 360°, less than 360° + 90°  So, it will be in  1st quadrant. Since sin is positive in 4th quadrant So, sign will be positive ∴ sin (2π + x) = sin x We can also prove it by We know that sin repeats after 2π ⇒ sin (2π + x) = sin (x) cos (2π + x) Since it is 2π, cos will remain cos Here x is an acute angle So, 2π + x = 360 + x 360 – x is an angle which is greater than 360°, less than 360° + 90°  So, it will be in  1st quadrant. Since cos is positive in 4th quadrant So, sign will be positive ∴ cos (2π + x) = cos x We can also prove it by We know that cos repeats after 2π ⇒ cos (2π + x) = cos (x)

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
Jail