Express Sine in Terms of Secant or Cosecant

By Mary Jane Sterling

Even though each trig function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of secant or cosecant, so that all the functions match, making it easier to solve the equation.

The easiest way to do this is to start with the Pythagorean identity,


solve for the sine in terms of cosine and replace each cosine with 1 over its reciprocal (which is secant):


The radical has a fraction in it. A better form is to simplify that fraction, so find a common denominator and split the fraction into two radicals — the bottom one of which you can further simplify:


The last function to write sine in terms of is the cosecant. The reciprocal of cosecant is sine, so this equation is just one of the basic reciprocal identities: