How to Convert a Tangent to Secant

Use the inverse tangent funtion to find an unknown angle if you know its tangent.

If you have a triangle with one right (90 degree) angle, then you can use elementary trigonometric functions to determine relationships between the remaining angles and sides of the triangle. The length of the side opposite one of the non-right angles divided by the length of the adjacent side will be the tangent of that angle. The length of the hypotenuse divided by the length of the adjacent side will be the secant of the angle. If you know the tangent function of an unknown angle, it's a simple matter to determine the secant function of that angle using a scientific calculator.

Things You'll Need

• Scientific calculator

Instructions

1. Ensure your calculator is working with the desired system of angle representation. Modern scientific calculators are usually able to work with degrees, radians or gradians. If you don't know what radians and gradians are, that's okay. Your calculator probably defaults to degrees. If you'd rather work with radians or gradians, that's okay too. Just make sure that you use the same system throughout your calculations.

2. Use the inverse tangent function to find the angle that corresponds to the tangent, which you know. Your calculator may have a separate button for "inverse." You'll need to press this button first and then "tan" to take the inverse tangent. Alternately, your calculator may have a button specifically for inverse tangent labeled "tan^-1."

3. Avoid confusing tan^-1 for one over the tangent (i.e., the reciprocal of the tangent). That is the cotangent function and entirely different from the inverse tangent. The inverse of any trigonometric function (sin^-1, cos^-1, etc) means finding the angle that would return the given number if you were to perform the trigonometric function on the angle. Tan^-1(x) means finding out what angle has a tangent equal to x.

4. Take the angle that you obtained from the inverse tangent function and find out what the cosine of that angle is. You must find the cosine, as your calculator probably doesn't have a secant button. The relationship between cosine and secant is straightforward.

5. Take the reciprocal of the cosine to find the secant. Your calculator probably has a "1/x" button to make this easier. The reciprocal of the cosine equals the secant.

Tips & Warnings

• It's a good idea to remember what all the trigonometric functions mean. In a right triangle, the hypotenuse is the side opposite the right angle. Each of the two remaining angles will be have the hypotenuse on one side and the adjacent side on the other, with the opposite side across the triangle from it. The following trigonometric functions describe the relationships between the triangle's sides:

• sine = opposite/hypotenuse

• cosine = adjacent/hypotenuse

• tangent = opposite/adjacent

• cotangent = adjacent/opposite

• secant = hypotenuse/adjacent

• cosecant=hypotenuse/opposite

References

• S.O.S. Math; Trigonometric Functions; Luis Valdez Sanchez; 1996
• Interactive Mathematics; Sine, Cosine, Tangent and the Reciprocal Ratios
• Hofstra University; The Trigonometric Functions; Stefan Waner and Steven R. Costenoble; 1997