Reciprocal vs. Inverse trig

Reciprocal Trig Functions

The normal trigonometric functions are sine, cosine, and tangent. However, there are also expressions for the reciprocals of these functions.

These functions are secant, cosecant, and cotangent. They are equal to the reciprocals of the ratios from SOHCAHTOA:

Secant = 1Cosine=HypotenuseAdjacent

Cosecant =1Sine=HypotenuseOpposite

Cotangent=1Tangent=AdjacentOpposite

The easiest way to remember this is that for sine and cosine, each pair should have one S and one C. So Secant and Cosine are paired and Cosecant and Sine are paired. Cotangent and tangent are generally much easier to remember, as they are the only identities that have a T in them.

How It Appears on the Exam

The reciprocal trig functions are generally used in right triangle problems. This means they are used just like in a normal SOHCAHTOA problem, but where the ratios are inverted.

More rare problems will involve looking at graphs of these functions. The important thing to remember is that just like their functions, their graphs will also be the reciprocals of the sine and cosine graphs (cotangent graphs are not likely to come up).

Sine there are points where sine and cosine are = 0, these graphs have vertical asymptotes at those points. This is why the graphs of these functions look like this:

Inverse Trig Functions

Inverse trig functions are how to calculate angles from the SOHCAHTOA ratios

• They are written using the normal trig functions, but with negative exponents: sin⁻¹, cos⁻¹, tan⁻¹

Since 

sin(θ)=OppositeHypotenuse, 

sin−1(OppositeHypotenuse)=θ, where θis the angle of the triangle with those opposite and hypotenuse sides.

Problems using the inverse trig functions generally will not ask a student to actually calculate a numerical value for the angle. Instead, students will simply be asked to understand how the inverse trig functions are applied to find an angle measure.