Trigonometric Ratios of Standard Angles in First and Second Quadrants

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  • A circle with the centre at the origin and radius 1 is a standard unit circle. Let P(x, y) be any point on the unit circle with m∠ XOP = θ. Now P lies on the unit circle. Hence l(OP) = 1.

Let PM be perpendicular to OX. Thus ΔOMP is right angled triangle.

By Pythagoras theorem

OM2 + MP2 = OP2



x2 + y2 = 1

Then by definition of trigonometric ratios

sin θ = length of opposite side / Length of hypotenuse = PM/ OP = y/1 = y

cos θ= length of adjacent side / Length of hypotenuse = OM/ OP = x/1 = x



tan θ = length of opposite side / length of adjacent side = PM/OM = y/x  (x not equal to 0)

cosec θ =  Length of hypotenuse / length of opposite side / = OP/ PM = 1/y (y not equal to 0)

sec θ =  Length of hypotenuse / length of adjacent side / = OP/ OM = 1/x (x not equal to 0)

cot θ = length of adjacent side / length of opposite side = OM/PM = x/y  (y not equal to 0)

From above values we can see that



cosec θ = 1/sin θ (if sin θ not equal to zero)

sec θ = 1/cos θ (if cos θ not equal to zero)

cot θ = 1/tan θ (if sin θ not equal to zero)

Notes:

  • The trigonometric functions do not depend on the position of the point P on the terminal ray but they depend on measure of angle q.
  • Coterminal angles have the same trigonometric functions
  • Since x = cos θ and y = sinθ, point P has coordinates (cos θ, sin θ).

Let P(x, y) be on the standard unit circle such that

x2 + y2 = 1



x2  ≤  1

– 1 ≤ x ≤ 1

– 1 ≤ cos θ ≤ 1

Similarly

y2  ≤  1



– 1 ≤ y ≤ 1

– 1 ≤ sin θ ≤ 1

Similarly

sec θ ≥ 1 or sec θ ≤ – 1



cosec θ ≥ 1 or cosec θ ≤ – 1

tan θ and cot θ can be any real numbers.

Trigonometric Ratios of 0o or 0c:

Let m∠ AOP = θ = 0o = 0c

Ray OA is the initial arm of the angle.

The terminal arm of the angle ray OP intersects the circle at P(1, 0)



Hence x = 1 an y = 0. Thus

sin 0o = y = 0

cos 0o =  x = 1

tan 0o =  y/x  = 0/1 = 0



cosec 0o = 1/y (Not defined since y = 0)

sec 0o  =  1/x = 1/1 = 1

cot 0o = x/y (Not defined since y = 0)

sin 0o

sin (0)c

cos 0o

cos (0)c

tan 0o



tan (0)c

cosec 0o

cosec (0)c

sec 0o

sec (0)c

cot 0o

cot (0)c

0 1 0 1

Trigonometric Ratios of 30o or (π/6)c:

Let m∠ AOP = θ = 30o = (π/6)c

Ray OA is the initial arm of the angle.

The terminal arm of the angle ray OP intersects the circle at P(x, y)

Let PM be perpendicular to OX. Thus ΔOMP is 30o-60o-90o triangle

PM = 1/2(OP) = 1/2 (1) = 1/2  (side opposite to 30o)

OM = 3/2(OP) = 3/2 (1) = 3/2  (side opposite to 60o)

Point P is in the first quadrant

Hence x = 3/2 an y = 1/2. Thus

sin 30o = y = 1/2

cos 30o =  x = 3/2

tan 30o =  y/x  = (1/2)/(3/2) = 1/3

cosec 30o = 1/y = 1/(1/2) = 2

sec 30o  =  1/x = 1/(3/2) = 2/3

cot 30o = x/y = (3/2)/(1/2) = 3

sin 30o

sin (π/6)c

cos 30o

cos (π/6)c

tan 30o

tan (π/6)c

cosec 30o

cosec (π/6)c

sec 30o

sec (π/6)c

cot 30o

cot (π/6)c

1/2 3/2 1/3 2 2/√3

3

Trigonometric Ratios of 45o or (π/4)c:

Let m∠ AOP = θ = 45o = (π/4)c

Ray OA is the initial arm of the angle.

The terminal arm of the angle ray OP intersects the circle at P(x, y)

Let PM be perpendicular to OX. Thus ΔOMP is 45o-45o-90o triangle

PM = 1/2(OP) = 1/2 (1) = 1/2  (side opposite to 45o)

OM = 1/2(OP) = 1/2 (1) = 1/√2  (side opposite to 45o)

Point P is in the first quadrant

Hence x = 1/2 an y = 1/2. Thus

sin 45o = y = 1/2

cos 45o =  x = √1/√2

tan 45o =  y/x  = (1/2)/(1/2) = 1

cosec 45o = 1/y = 1/(1/2) = 2

sec 45o  =  1/x = 1/(1/2) = 2

cot 45o = x/y = (1/2)/(1/2) = 1

sin 45o

sin (π/4)c

cos 45o

cos (π/4)c

tan 45o

tan (π/4)c

cosec 45o

cosec (π/4)c

sec 45o

sec (π/4)c

cot 45o

cot (π/4)c

1/2 1/√2 1 2 2

1

Trigonometric Ratios of 60o or (π/3)c:

Let m∠ AOP = θ = 60o = (π/3)c

Let PM be perpendicular to OX. Thus ΔOMP is 30o-60o-90o triangle

PM = 3/2(OP) = 1/2 (1) = 3/2  (side opposite to 60o)

OM = 1/2(OP) = 3/2 (1) = 1/2  (side opposite to 30o)

Point P is in the first quadrant

Hence x = 1/2 an y = 3/2. Thus

sin 60o = y = 3/2

cos 60o =  x = 1/2

tan 60o =  y/x  = (3/2)/(1/2) = 3

cosec 60o = 1/y = 1/(3/2) = 2/3

sec 60o  =  1/x = 1/(1/2) = 2

cot 60o = x/y = (1/2)/(3/2) = 1/3

sin 60o

sin (π/3)c

cos 60o

cos (π/3)c

tan 60o

tan (π/3)c

cosec 60o

cosec (π/3)c

sec 60o

sec (π/3)c

cot 60o

cot (π/3)c

3/2 1/2 3 2/3 2

1/√3

Trigonometric Ratios of 90o or (π/2)c:

Let m∠ AOP = θ = 90o = (π/2)c

Let PM be perpendicular to OX. M coicides with O

PM = 1  and OM = 0

Point P is on postive y-axis

Hence x = 0 an y = 1. Thus

sin 90o = y = 1

cos 90o =  x = 0

tan 90o =  y/x  (Not defined since x = 0)

cosec 90o = 1/y = 1/1 = 1

sec 90o  =  1/x  (Not defined since x = 0)

cot 90o = x/y = 0/1 = 0

sin 90o

sin (π/2)c

cos 90o

cos (π/2)c

tan 90o

tan (π/2)c

cosec 90o

cosec (π/2)c

sec 90o

sec (π/2)c

cot 90o

cot (π/2)c

1 0 1

0

Trigonometric Ratios of 120o or (2π/3)c:

Let m∠ AOP = θ = 120o = (2π/3)c

m∠ POM = 60o

Let PM be perpendicular to OX’. Thus ΔOMP is 30o-60o-90o triangle

PM = 3/2(OP) = 1/2 (1) = 3/2  (side opposite to 60o)

OM = 1/2(OP) = 3/2 (1) = 1/2  (side opposite to 30o)

Point P is in the second quadrant

Hence x = – 1/2 an y = 3/2. Thus

sin 120o = y = 3/2

cos 120o =  x = –1/2

tan 120o =  y/x  = (3/2)/(-1/2) = –3

cosec 120o = 1/y = 1/(3/2) = 2/3

sec 120o  =  1/x = 1/(-1/2) = – 2

cot 120o = x/y = (-1/2)/(3/2) = – 1/3

sin 120o

sin (2π/3)c

cos 120o

cos (2π/3)c

tan 120o

tan (2π/3)c

cosec 120o

cosec (2π/3)c

sec 120o

sec (2π/3)c

cot 120o

cot (2π/3)c

3/2 -1/2 -√3 2/3 -2

-1/√3

Trigonometric Ratios of 135o or (3π/4)c:

Let m∠ AOP = θ = 135o = (3π/4)c

m∠ POM = 45o

Ray OA is the initial arm of the angle.

The terminal arm of the angle ray OP intersects the circle at P(x, y)

Let PM be perpendicular to OX’. Thus ΔOMP is 45o-45o-90o triangle

PM = 1/2(OP) = 1/2 (1) = 1/2  (side opposite to 45o)

OM = 1/2(OP) = 1/2 (1) = √1/√2  (side opposite to 45o)

Point P is in the second quadrant

Hence x = – 1/2 an y = 1/2. Thus

sin 135o = y = 1/2

cos 135o =  x = – 1/√2

tan 135o =  y/x  = (-1/2)/(1/2) = – 1

cosec 135o = 1/y = 1/(1/2) = 2

sec 135o  =  1/x = 1/(-1/2) = – 2

cot 135o = x/y = (-1/2)/(1/2) = – 1

sin 135o

sin (3π/4)c

cos 135o

cos (3π/4)c

tan 135o

tan (3π/4)c

cosec 135o

cosec (3π/4)c

sec 135o

sec (3π/4)c

cot 135o

cot (3π/4)c

1/2 – 1/√2 – 1 2 – √2

– 1

Trigonometric Ratios of 150o or (5π/3)c:

Let m∠ AOP = θ = 150o = (5π/6)c

m∠ POM = 30o

Ray OA is the initial arm of the angle.

The terminal arm of the angle ray OP intersects the circle at P(x, y)

Let PM be perpendicular to OX’. Thus ΔOMP is 30o-60o-90o triangle

PM = 1/2(OP) = 1/2 (1) = 1/2  (side opposite to 30o)

OM = 3/2(OP) = 3/2 (1) = 3/2  (side opposite to 60o)

Point P is in the second quadrant

Hence x = – 3/2 an y = 1/2. Thus

sin 150o = y = 1/2

cos 150o =  x = – 3/2

tan 150o =  y/x  = (1/2)/(-3/2) = – 1/3

cosec 150o = 1/y = 1/(1/2) = 2

sec 150o  =  1/x = 1/(-3/2) = – 2/3

cot 150o = x/y = (-3/2)/(1/2) = – 3

sin 150o

sin (5π/6)c

cos 150o

cos (5π/6)c

tan 150o

tan (5π/6)c

cosec 150o

cosec (5π/6)c

sec 150o

sec (5π/6)c

cot 150o

cot (5π/6)c

1/2 – √3/2 – 1/3 2 – 2/√3

– √3

Trigonometric Ratios of 180o or πc:

Trigonometric Ratios

Let m∠ AOP = θ = 180o = πc

Ray OA is the initial arm of the angle.

The terminal arm of the angle ray OP intersects the circle at P(-1, 0)

Hence x = -1 an y = 0. Thus

sin 180o = y = 0

cos 180o =  x = – 1

tan 180o =  y/x  = 0/-1 = 0

cosec180o = 1/y (Not defined since y = 0)

sec 180o  = 1/x = 1/-1 = – 1

cot 180o = x/y (Not defined since y = 0)

sin 180o

sin (π)c

cos 180o

cos (π)c

tan 180o

tan (π)c

cosec 180o

cosec (π)c

sec 180o

sec (π)c

cot 180o

cot (π)c

0 – 1 0 – 1

 

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