Which trigonometric function is equivalent to the expression

The graph of an odd function is symmetric about the origin. The other even-odd identities follow from the even and odd nature of the sine and cosine functions. We can interpret the tangent of a negative angle as. We can interpret the cotangent of a negative angle as. The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as.

Finally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as. To sum up, only two of the trigonometric functions, cosine and secant, are even.

The other four functions are odd, verifying the even-odd identities. We first encountered these identities in Section 5. The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions. Given a trigonometric identity, verify that it is true.

There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side. In the first method, we split the fraction, putting both terms in the numerator over a common denominator. This problem illustrates that there are multiple ways we can verify an identity. Employing some creativity can sometimes simplify a procedure. As long as the substitutions are correct, the answer will be the same. Use the sum of angles identities or double angle identities to break apart any sums of angles or to replace double angles.

If you end up with a fraction on one side of the identity but not the other then multiply the non-fraction side by a UFOO to convert into a fraction. A UFOO is a fraction that equals 1 because it has equal numerator and denominator. Example 3 requires one. When the two sides are identical the identity is proven. Algebra Coach Exercises. Co-function Identities for Angles in the First Quadrant:.

Co-function Identities for Angles in the Second Quadrant:. Using Right Angle to Confirm:. Unknown 4 May at Anonymous 4 May at