Trigonometric Identities & 7.2 Trigonometric Integrals

Pythagorean Identity

$$\sin^2x+\cos^2x=1$$

$\sin^2x+\cos^2x=1$ transformed for $\sin^2x$ ($\sin^2x=?$)

$$\sin^2x=1-\cos^2x$$

$\sin^2x+\cos^2x=1$ transformed for $\cos^2x$ ($\cos^2x=?$)

$$\cos^2x=1-\sin^2x$$

$\sin^2x+\cos^2x=1$ transformed for $\csc^2x$ ($\csc^2x=?$ )

$\csc^2x=1+\cot^2x$ (divide original pythagorean identity by $\sin^2x$)

$\sin^2x+\cos^2x=1$transformed for $\cot^2x$ ($\cot^2x=?$)

$\cot^2x=\csc^2x-1$(divide original pythagorean identity by $\sin^2x$ )

$\sin^2x+\cos^2x=1$ transformed for $\tan^2x$ ($\tan^2x=?$)

$\tan^2x=\sec^2x-1$ (divide original pythagorean identity by $\cos^2x$)

$\sin^2x+\cos^2x=1$ transformed for $\sec^2x$($\sec^2x=?$)

$\sec^2x=\tan^2x+1$ (divide original pythagorean identity by $\cos^2x$)

double angle identity of $\sin\left(2x\right)$ ($\sin\left(2x\right)=?$ )

$$\sin\left(2x\right)=2\sin x\cos x$$

double angle identity of $\cos\left(2x\right)$ ($\cos\left(2x\right)=?$ )

$$\cos\left(2x\right)=\cos^2x-\sin^2x=2\cos^2x-1=1-\sin^2x$$

half angle identity of $\cos^2x$ ($\cos^2x=?$)

$$\cos^2x=\frac{1+\cos\left(2x\right)}{2}$$

half angle identity of $\sin^2x$ ($\sin^2x=?$)

$$\sin^2x=\frac{1-\cos\left(2x\right)}{2}$$

the degree of sine m is an odd integer in $\int\sin^{m}x\cos^{n}xdx$, with n as an integer > 0

1. reserve a power of sine (e.g., $\int\sin^3x\cos^4xdx=\int\sin^2x\cos^4x\sin xdx$)

2. convert the remaining powers of sine to cosine using trigonometric identities

3. perform a u-substitution of cosine

4. solve

the degree of cosine n is an odd integer in $\int\sin^{m}x\cos^{n}xdx$, with m as an integer > 0

1. reserve a power of cosine (e.g., $\int\sin^4x\cos^3xdx=\int\sin^4x\cos^2x\cos xdx$ )

2. convert the remaining powers of cosine to sine using trigonometric identities

3. perform a u-substitution of sine

4. solve

the degree of both sine m and cosine n are odd in $\int\sin^{m}x\cos^{n}xdx$

(pick either method)

1. reserve a power of either cosine or sine

2. convert the remaining powers of the chosen function to the other function using trigonometric identities

3. perform a u-substitution of the other function

4. solve

he degree of both sine m and cosine n are even in $\int\sin^{m}x\cos^{n}xdx$

1. substitute both sine and cosine with their half-angle identities

2. distribute and solve

the degree of tangent m is an odd integer in $\int\tan^{m}x\sec^{n}xdx$, with n as an integer > 0

1. reserve a $\sec x\tan x$(e.g., $\int\tan^3x\sec xdx=\int\tan^2x\sec x\tan xdx$ )

2. convert the remaining powers of tangent to secant using trigonometric identities

3. perform a u-substitution of secant

4. solve

the degree of secant n is an even integer in $\int\tan^{m}x\sec^{n}xdx$, with m as an integer > 0

1. reserve a $\sec^2x$ (e.g., $\int\tan^6x\sec^6xdx=\int\tan^6x\sec^4x\sec^2xdx$ )

2. convert the remaining powers of secant to tangent using trigonometric identities

3. perform a u-substitution of tangent

4. solve