DE-BOOK Trigonometric Identities

Section Trigonometric Identities

Trigonometric identities appear frequently in differential equations, especially when working with oscillatory solutions like \(y(t) = A\cos(\omega t) + B\sin(\omega t)\text{.}\) Knowing a few core identities helps simplify expressions, evaluate integrals, and recognize equivalent forms—whether you’re solving second-order linear equations or analyzing Laplace transforms.

🔗

✳️ Pythagorean Identities.

These identities are derived from the unit circle and are often used to simplify squared trigonometric terms. They are especially useful when converting between forms or integrating expressions:

🔗

\begin{equation*} 1.\quad \sin^2(\theta) + \cos^2(\theta) = 1 \end{equation*}

🔗

\begin{equation*} {\small\DLGb\textit{divide both sides by}\ \cos^2(\theta)}\qquad\swarrow \end{equation*}

🔗

\begin{equation*} \searrow\qquad{\DLGb\small\textit{divide both sides by}\ \sin^2(\theta)} \end{equation*}

🔗

\begin{equation*} 2.\quad \tan^2(\theta) + 1 = \sec^2(\theta) \end{equation*}

🔗

\begin{equation*} \end{equation*}

🔗

\begin{equation*} 3.\quad 1 + \cot^2(\theta) = \csc^2(\theta) \end{equation*}

🔗

✳️ Even and Odd Properties.

These identities help you simplify expressions involving negative angles. They’re especially important when analyzing symmetry in solutions or applying inverse Laplace transforms:

🔗

\(1.\ \ \sin(-\theta) = -\sin(\theta)\)

🔗

\(4.\ \ \csc(-\theta) = -\csc(\theta)\)

🔗

\(2.\ \ \cos(-\theta) = \cos(\theta)\)

🔗

\(5.\ \ \sec(-\theta) = \sec(\theta)\)

🔗

\(3.\ \ \tan(-\theta) = -\tan(\theta)\)

🔗

\(6.\ \ \cot(-\theta) = -\cot(\theta)\)

🔗

You have attempted of activities on this page.

🔗

Prev Top Next