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Reference

Trigonometric Ratios

Concept

Trigonometric Ratios

A trigonometric ratio relates two side lengths of a right triangle. Consider the right triangle One of its acute angles has been named

Since it is opposite to the right angle, is the hypotenuse of the right triangle. The remaining sides — the legs — can be named relative to the marked angle Because is next to it is called the adjacent side. Conversely, because lies across from it is called the opposite side.

Trigonometric ratios can be used to determine unknown side lengths or angle measures in right triangles.

Concept

Sine

For an acute angle of a right triangle, the sine of is the ratio between the lengths of the opposite side and the hypotenuse.
Sides of a right triangle labeled
The sine of is written as

This trigonometric ratio states the ratio between the opposite side and the hypotenuse. It gives no indication about the lengths of the individual sides.
Right triangle with two vertices movable and the sine of one acute angle is computed while the measures change
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.

Concept

Cosine

For an acute angle of a right triangle, the cosine of is the ratio between the lengths of the adjacent side and the hypotenuse.
Sides of a right triangle labeled
The cosine of is written as

While this trigonometric ratio expresses the ratio of the adjacent side to the hypotenuse, it does not state the actual measurements of their lengths.
Right triangle with two vertices movable and the cosine of one acute angle is computed while the measures change
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.

Concept

Tangent

For an acute angle of a right triangle, the tangent of is the ratio between the lengths of the opposite side and the adjacent side.
Sides of a right triangle labeled
The tangent of is written as

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.
Right triangle with two vertices movable and the tangent of one acute angle is computed while the measures change
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.

Concept

Cosecant

For an acute angle of a right triangle, the cosecant of is the ratio between the lengths of the hypotenuse and the opposite side.
Sides of a right triangle labeled
The cosecant of is written as

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.
Right triangle with two vertices movable and the cosecant of one acute angle is computed while the measures change
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 is the reciprocal of the sine of

Concept

Secant

For an acute angle of a right triangle, the secant of is the ratio between the lengths of the hypotenuse and the adjacent side.
Sides of a right triangle labeled
The secant of is written as

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.
Right triangle with two vertices movable and the secant of one acute angle is computed while the measures change
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 is the reciprocal of the cosine of

Concept

Cotangent

For an acute angle of a right triangle, the cotangent of is the ratio between the lengths of the adjacent side and the opposite side.
Sides of a right triangle labeled
The cotangent of is written as

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.
Right triangle with two vertices movable and the cotangent of one acute angle is computed while the measures change
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 is also the reciprocal of the tangent of