Sine, cosine, and tangent are the more commonly used trigonometric ratios, although you should also learn cosecant, secant, and cotangent to have an in-depth knowledge of the trigonometric table.

For example, for the first entry in the sine column (sin 0°), set x to equal 0 and plug it into the expression √x/2. This will give you √0/2, which can be simplified to 0/2 and then finally to 0. Plugging the angles into the expression √x/2 in this way, the remaining entries in the sine column are √1/2 (which can be simplified to ½, since the square root of 1 is 1), √2/2 (which can be simplified to 1/√2, since √2/2 is also equal to (1 x √2)/(√2 x √2) and in this fraction, the “√2” in the numerator and a “√2” in the denominator cancel each other out, leaving 1/√2), √3/2, and √4/2 (which can be simplified to 1, since the square root of 4 is 2 and 2/2 = 1). Once the sine column is filled, it’ll be a lot easier to fill in the remaining columns.

For example, since 1 is the value placed in the final entry in the sine column (sine of 90°), this value will be placed in the first entry for the cosine column (cosine of 0°). Once filled, the values in the cosine column should be 1, √3/2, 1/√2, ½, and 0.

To take 30° as an example: tan 30° = sin 30° / cos 30° = (√1/2) / (√3/2) = 1/√3. The entries of your tangent column should be 0, 1/√3, 1, √3, and undefined for 90°. The tangent of 90° is undefined because sin 90° / cos 90° = 1/0 and division by 0 is always undefined.

For instance, use the sine of 90° to fill in the entry for the cosecant of 0°, the sine of 60° for the cosecant of 30°, and so on.

This is mathematically valid because the inverse of the cosine of an angle is equal to that angle’s secant.

This works because the cotangent of an angle is equal to the inversion of the tangent of an angle. You can also find the cotangent of an angle by dividing its cosine by its sine.

If you’re calculating sine, cosine, or tangent in the context of a math class, it’s likely you’ll already be working with a right triangle.

The sine of an angle is equal to the opposite side divided by the hypotenuse. The cosine of an angle is equal to the adjacent side divided by the hypotenuse. Finally, the tangent of an angle is equal to the opposite side divided by the adjacent side. For example, to determine the sine of a 35°, you would divide the length of the opposite side of the triangle by the hypotenuse. If the opposite side’s length was 2. 8 and the hypotenuse was 4. 9, then the sine of the angle would be 2. 8/4. 9, which is equal to 0. 57.

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Thus, because cosecant is the inverse of sine, it is equal to the hypotenuse divided by the opposite side. The secant of an angle is equal to the hypotenuse divided by the adjacent side. The cotangent of an angle is equal to the adjacent side divided by the opposite side. For example, if you wanted to find the cosecant of a 35°, with an opposite side length of 2. 8 and a hypotenuse of 4. 9, you would divide 4. 9 by 2. 8 to get a cosecant of 1. 75.