Trigonometric Ratios

For an acute angle θ of a right triangle, the sine of θ is the ratio between the lengths of the opposite side and the hypotenuse.

The sine of θ is written as sin θ.

This trigonometric ratio states the ratio between the opposite side and the hypotenuse. It gives no indication about the lengths of the individual sides.

It should be noted that for a given angle, when the opposite side and hypotenuse measurements change, if their ratio stays the same, then the sine, too, would stay the same.

For an acute angle θ of a right triangle, the cosine of θ is the ratio between the lengths of the adjacent side and the hypotenuse.

The cosine of θ is written as cos θ.

While this trigonometric ratio expresses the ratio of the adjacent side to the hypotenuse, it does not state the actual measurements of their lengths.

It should be noted that for a given angle, when the adjacent side and hypotenuse lengths change, if their ratio stays the same, then the cosine, too, would stay the same.

For an acute angle θ of a right triangle, the tangent of θ is the ratio between the lengths of the opposite side and the adjacent side.

The tangent of θ is written as tan θ.

The trigonometric ratio expresses the ratio between the opposite side and the adjacent side. It does not give any indication about the actual measurement of the side lengths.

It should be noted that for a given angle, when the adjacent side and the opposite side lengths change, if their ratio stays the same, then the tangent, too, would stay the same.

For an acute angle θ of a right triangle, the cosecant of θ is the ratio between the lengths of the hypotenuse and the opposite side.

The cosecant of θ is written as csc(θ).

This trigonometric ratio states the ratio between the hypotenuse and the opposite side to a certain angle. It gives no indication about the lengths of the sides.

When the opposite side and hypotenuse lengths change for a given angle, if their ratio stays the same then the cosecant, too, would stay the same. Also note that the cosecant of θ is the reciprocal of the sine of θ.

csc θ=hyp/opp sin θ =opp/hyp ⇒ csc θ = 1/sin θ

For an acute angle θ of a right triangle, the secant of θ is the ratio between the lengths of the hypotenuse and the adjacent side.

The secant of θ is written as sec(θ).

This trigonometric ratio states the ratio between the hypotenuse and the adjacent side to a certain angle. It gives no indication about the lengths of the sides.

When the adjacent side and hypotenuse lengths change for a given angle, if their ratio stays the same then the secant, too, would stay the same. Also note that the secant of θ is the reciprocal of the cosine of θ.

sec θ=hyp/adj cos θ =adj/hyp ⇒ sec θ = 1/cos θ

For an acute angle θ of a right triangle, the cotangent of θ is the ratio between the lengths of the adjacent side and the opposite side.

The cotangent of θ is written as cot θ.

This trigonometric ratio states the ratio between the adjacent side to a certain angle and the opposite side to the angle. It gives no indication about the lengths of the sides.

When the adjacent side and the opposite side lengths for a given angle change, if their ratio stays the same, then the cotangent, too, would stay the same. In addition, note that the cotangent of θ is also the reciprocal of the tangent of θ.

cot θ=adj/opp tan θ =opp/adj ⇒ cot θ = 1/tan θ

Adjust the trigonometric ratio and the angle to see the resulting value.

Table of Values

θ		sin θ	cos θ	tan θ
Degrees	Radians
0	0	0	1	0
30	π/6	1/2	sqrt(3)/2	sqrt(3)/3
45	π/4	sqrt(2)/2	sqrt(2)/2	1
60	π/3	sqrt(3)/2	1/2	sqrt(3)
90	π/2	1	0	undefined
120	2π/3	sqrt(3)/2	-1/2	-sqrt(3)
135	3π/4	sqrt(2)/2	-sqrt(2)/2	-1
150	5π/6	1/2	-sqrt(3)/2	-sqrt(3)/3
180	π	0	-1	0
210	7π/6	-1/2	-sqrt(3)/2	sqrt(3)/3
225	5π/4	-sqrt(2)/2	-sqrt(2)/2	1
240	4π/3	-sqrt(3)/2	-1/2	sqrt(3)
270	3π/2	-1	0	undefined
300	5π/3	-sqrt(3)/2	1/2	-sqrt(3)
315	7π/4	-sqrt(2)/2	sqrt(2)/2	-1
330	11π/6	-1/2	sqrt(3)/2	-sqrt(3)/3
360	2π	0	1	0