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Topics You Need To Understand For This Page

Trig Identities and Formulas

basic trig identities

\(\sin^2\theta+\cos^2\theta=1\)   |   \(1+\tan^2\theta=\sec^2\theta\)

\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\)   |   \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\)

\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\)   |   \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\)

power reduction (half-angle) formulae

\(\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}\)   |   \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\)

double angle formulae

\(\sin(2\theta)=2\sin\theta\cos\theta\)   |   \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\)

links

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Trig Derivatives and Integrals

basic trig derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = }\) \(\sec(t)\tan(t) \)

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = }\) \( -\csc(t)\cot(t) \)

basic trig integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \ln\abs{\sec(x)+\tan(x)}+C\)

\(\int{\csc(x)~dx} = -\ln\abs{\csc(x)+\cot(x)}+C\)

reduction formulae

Reduction Formulas (n is a positive integer)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\sin^{n-2}x~dx} }\)

\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\)

Reduction Formulas (n is an integer and \(n>1\))

\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}\)

\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1} + }\) \(\displaystyle{ \frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\)

links

17calculus trig formulas - full list

Related Topics and Links

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

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calculus motivation - music and learning

This page covers the derivation and use of the sine reduction formula for integration.

Sine Reduction Formula (where n is a positive integer)

\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int{\sin^{n-2}x~dx}}\)

When you have an integral with only sine where the power is greater than one, you can use the sine reduction formula, repeatedly if necessary, to reduce the power until you end up with either \(\sin x\) or \(\sin^2 x\). Let's derive the formula and then work some practice problems.

Deriving The Sine Reduction Formula

First, we separate out one \(\sin(x)\) term.

\( \int{\sin^nx~dx} = \int{\sin^{n-1}x\sin x~dx} \)

Now we use integration by parts

\(u=\sin^{n-1}x ~~~ \to \) \( ~~~ du=(n-1)\sin^{n-2}x\cos x~dx\)

\(dv=\sin x~dx ~~~ \to \) \( ~~~ v = -\cos x\)

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x - \) \(\int{-\cos x(n-1)\sin^{n-2}x\cos x~dx}\)

Simplify

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x\cos^2x~dx}\)

Now we use the identity \(\cos^2x+\sin^2x=1\) to replace \(\cos^2x\) with \(1-\sin^2x\) in the last integral.

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~(1-\sin^2x)~dx}\)

Next distribute the \(\sin^{n-2}x\).

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x-\sin^nx~dx}\)

Separate the integral on the right into two integrals. Don't forget that the \((n-1)\) factor needs to be applied to both integrals.

\(\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~dx} - \) \((n-1)\int{\sin^nx~dx}\)

Now we add \((n-1)\int{\sin^nx~dx} \) to both sides of the equal sign.

\(\int{\sin^nx~dx} + (n-1)\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~dx} \)

Factor \(\int{\sin^nx~dx}\) on the left and notice that \(1+n-1 = n\).

\(n\int{\sin^nx~dx} = \) \(-\cos x\sin^{n-1}x \) + \((n-1)\int{\sin^{n-2}x~dx} \)

Divide both sides by \(n\) to solve for \(\int{\sin^nx~dx}\).

\(\displaystyle{\int{\sin^nx~dx} = }\) \(\displaystyle{\frac{-\cos x\sin^{n-1}x}{n} }\) + \(\displaystyle{\frac{n-1}{n}\int{\sin^{n-2}x~dx} }\)

This last equation is the sine reduction formula. Here is a video showing this derivation in a similar way.

Dr Chris Tisdell - Reduction formula \(\sin^n x\) [5min-4secs]

video by Dr Chris Tisdell

Now let's work some practice problems.

Practice

Integrate \(\int{ \sin^5 x ~dx }\) using the reduction formula.

Problem Statement

Integrate \(\int{ \sin^5 x ~dx }\) using the reduction formula.

Solution

2300 solution video

video by PatrickJMT

close solution

\(\displaystyle{ \int_{0}^{\pi/2}{ \sin^7x ~dx } }\)

Problem Statement

\(\displaystyle{ \int_{0}^{\pi/2}{ \sin^7x ~dx } }\)

Solution

123 solution video

video by Dr Chris Tisdell

close solution
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