What Sine, Cosine, Tangent, and Cotangent Are All About

In mathematics, trigonometric functions are used to determine the acute angles of a right triangle. For that, the relationship of its sides, catheti, and hypotenuse are calculated.

The hypotenuse is the longest side of a right triangle — opposite the right angle. Catheti are the 2 other sides that make up the 90°- degree angle together. The cathetus, located next to the acute angle that should be calculated is called an adjacent leg, the one that is located opposite is called the opposite leg.

The trigonometric functions are used in many scientific spheres. For example, in astronomy for calculating the location of celestial bodies or in surveying — to calculate lengths, areas, and relative angles between objects.

5-Minute Crafts wants to tell you about the most widely used trigonometric functions (sine, cosine, and tangent) and their reciprocals (cosecant, secant, and cotangent).

1. Sine

The sine of an acute angle of a right triangle is equal to the ratio of the opposite leg to the hypotenuse. For the AOB triangle, the formula is sin α = AB / OB.

2. Cosine

The cosine of an acute angle of a right triangle is equal to the ratio of the adjacent leg to the hypotenuse. For the AOB triangle the formula is cos α = OA / OB.

3. Tangent

The tangent of an acute angle of a right triangle is equal to the ratio of the opposite leg to the adjacent one. For the AOB triangle, the formula is tg α = AB / OA.

The tangent is a derivative function that can also be calculated by dividing the sine by the cosine. The formula is tg α = sin α / cos α.

4. Cotangent

The cotangent of an acute angle of a right triangle is equal to the ratio of the adjacent leg to the opposite one. For the AOB triangle, the formula is ctg α = OA / AB.

Cotangent is a derived function that can also be calculated by dividing the cosine by the sine. The formula is ctg α = cos α / sin α.

5. Secant

The secant of an acute angle of a right triangle is equal to the ratio of the hypotenuse to the adjacent leg. For the AOB triangle, the formula is sec α = ОВ / OА.

The secant is a derived function that can also be calculated by dividing one by the cosine of an acute angle. The formula is: sec α = 1 / cos α.

6. Cosecant


The cosecant of an acute angle of a right triangle is equal to the ratio of the hypotenuse to the opposite leg. For the AOB triangle, the formula is cosec α = OB / AB.

The cosecant is a derived function that can also be calculated by dividing one by the sine of an acute angle. The formula is cosec α = 1 / sin α.

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