Tag Archives: Trigonometric Functions

Integration of Trigonometric Functions

Integration of Trigonometric Functions

Integration of $\cos{x}$ $$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \sin{x} &= \int{\cos{x}}dx \\ \therefore \int{\cos{x}}dx &= \sin{x} +c \\ \end{align} $$ Integration of $\cos{(ax+b)}$ $$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{(ax+b)} &= \cos{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\ &= \cos{(ax+b)} \times a \\ &= a\cos{(ax+b)} \\ \sin{(ax+b)} &= \int{a\cos{(ax+b)}}dx \\ &= a\int{\cos{(ax+b)}}dx \\ \dfrac{1}{a}\sin{(ax+b)} &= \int{\cos{(ax+b)}}dx \\ \therefore \int{\cos{(ax+b)}}dx […]

Derivative of Trigonometric Functions

Derivative of Trigonometric Functions

$$ \displaystyle \begin{align} \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \dfrac{d}{dx}\cos{x} &= -\sin{x} \\ \dfrac{d}{dx}\tan{x} &= \sec^2{x} \\ \end{align} $$ Example 1 Prove $\dfrac{d}{dx}\tan{x} = \sec^2{x}$ using $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\cos{x} = -\sin{x}$. \( \begin{align} \displaystyle \require{color} \dfrac{d}{dx}\tan{x} &= \dfrac{d}{dx}\dfrac{\sin{x}}{\cos{x}} \\ &= \dfrac{\dfrac{d}{dx}\sin{x} \times \cos{x}-\sin{x} \times \dfrac{d}{dx}\cos{x}}{\cos^2{x}} &\color{red} \text{quotient rule}\\ &= \dfrac{\cos{x} \times \cos{x}-\sin{x} \times (-\sin{x})}{\cos^2{x}} \\ […]

Integrating Trigonometric Functions by Double Angle Formula

Integrating Trigonometric Functions by Double Angle Formula

Integrating Trigonometric Functions by Double Angle Formula Integrating Trigonometric Functions can be done by Double Angle Formula reducing the power of trigonometric functions. \( \begin{aligned} \displaystyle \cos{2A} &= 2\cos^2{A} – 1 \\ &= 1 – 2\sin^2{A} \\ &= \cos^2{A} – \sin^2{A} \\ \end{aligned} \\ \) Practice Questions Question 1 Find \( \displaystyle \int{\cos^2{x}}dx \). \( […]