# Three each other. The derivative of sine

Three of the six trigonometric functions can be classified as complementary trigonometric functions. These functions are those trigonometric functions that start with the letters *co*: cosine, cotangent, and cosecant. Each cofunction is related to a trigonometric function that is not a cofunction based on complementary angles. For instance, a relationship exists the complementary angles of sine and cosine. Cofunctions are the only trigonometric functions whose derivative is negative. \$\$ frac{d}{dx} left cos{x}
ight = -sin{x} \$\$\$\$ frac{d}{dx} left cot{x}
ight = -csc^2{x} \$\$\$\$ frac{d}{dx} left csc{x}
ight = -csc{x} cot{x} \$\$Similarly, trigonometric functions that are not cofunctions are the only functions whose derivative is positive.\$\$ frac{d}{dx} left sin{x}
ight = cos{x} \$\$\$\$ frac{d}{dx} left an{x}
ight = sec^2{x} \$\$\$\$ frac{d}{dx} left sec{x}
ight = sec{x} an{x} \$\$Notice that sine and cosine are not quite derivatives of each other. The derivative of sine is cosine, but the derivative of cosine is negative sine.Use the product rule to determine the derivative of \(  sin{x} cos{x} ext{.} \)\$\$ egin{align} frac{d}{dx} left sin{x} cos{x}
ight &= frac{d}{dx} left sin{x}
ightcos{x} + sin{x} frac{d}{dx} left cos{x}
ight \ &= left( cos{x}
ight)cos{x} + sin{x} left( -sin{x}
ight) \&= cos^2{x} – sin^2{x} end{align}\$\$Once the derivatives of \( sin{x} \) and \( cos{x} \) are known, use the quotient rule, not the ??? rule ??? rule, to determine the derivatives of \( an{x} ext{,} \) \( cot{x} ext{,} \) \( sec{x} ext{,} \) and \( csc{x} ext{.} \) The quotient rule is used because each of the remaining trigonometric functions can be written as a quotient with \( sin{x} \) or \( cos{x} ext{.}\) For instance, \( an{x} = frac{sin{x}}{cos{x}} ext{.} \)