Tag Archives: Trigonometric Functions

Trigonometric Integration by Substitution

Trigonometric Integration by Substitution

Substitution of Angle Parts Example 1 Find $\displaystyle \int{(2x+3) \sin (x^2+3x)}dx$. \( \begin{align} \displaystyle\text{Let } u &= x^2+3x \\\dfrac{du}{dx} &= 2x + 3 \\du &= (2x+3)dx \\\int{(2x+3) \sin (x^2+3x)}dx &= \int{\sin u}du \\&= -\cos u + C \\&= -\cos {(x^2+3x)} + C \end{align} \) Substitution of $\sin{x}$ $$ \begin{align} \displaystyleu &= \sin{x} \\\dfrac{du}{dx} &= \cos{x} […]

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 $\cos{(ax+b)}$ $$ \begin{align} \displaystyle\dfrac{d}{dx}\sin{(ax+b)} &= \cos{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\&= \cos{(ax+b)} \times a \\&= a\cos{(ax+b)} \\\sin{(ax+b)} &= \int{a\cos{(ax+b)}}dx \\&= a\int{\cos{(ax+b)}}dx \\\dfrac{1}{a}\sin{(ax+b)} &= \int{\cos{(ax+b)}}dx \\\therefore \int{\cos{(ax+b)}}dx &= \dfrac{1}{a}\sin{(ax+b)} +c \end{align} $$ Example 1 Find $\displaystyle \int{\cos{(2x+4)}}dx$. \( […]

Derivative of Trigonometric Functions

Derivative of Trigonometric Functions

$$ \large \displaystyle \begin{align}\dfrac{d}{dx}\sin{x} &= \cos{x} \\\dfrac{d}{dx}\cos{x} &= -\sin{x} \\\dfrac{d}{dx}\tan{x} &= \sec^2{x} \\\end{align} $$ 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{AMSsymbols} \require{AMSsymbols} \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}} \\&= \dfrac{\cos^2{x} + \sin^2{x}}{\cos^2{x}} \\&= […]

Trigonometric Applications of Maximum and Minimum

Trigonometric Applications of Maximum and Minimum

Find the maximum value of $i=100\sin(50 \pi t +0.32)$, and the time when this maximum occurs. \( \begin{align} \displaystyle\dfrac{di}{dt} &= 0 \\100\cos(50 \pi t+ 0.32) \times 50 \pi &= 0 \\\cos(50 \pi t+ 0.32) &= 0 \\50 \pi t+ 0.32 &= \dfrac{\pi}{2}, \dfrac{3\pi}{2}, \cdots \\t &= \dfrac{1}{50 \pi}\Big(\dfrac{\pi}{2}-0.32\Big), \dfrac{1}{50 \pi}\Big(\dfrac{3\pi}{2}-0.32\Big) \cdots \\t &= 0.0080, 0.028, […]

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 \( \displaystyle \int{\cos^2{x}}dx \). \( \begin{aligned} \displaystyle2\cos^2{x}-1 &= \cos{2x} \\2\cos^2{x} &= \cos{2x} + 1 \\\cos^2{x} &= […]