Trigonometric Substitution

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.3 – Trigonometric Substitution

Trigonometric Substitution

Prerequisite Knowledge

Definition of Trig Functions on Right Triangle

• ${\displaystyle \cos (C)=\frac{\text{adjacent}}{\text{hypotenuse}}}$$\displaystyle \cos{(C)}=\frac{\text{adjacent}}{\text{hypotenuse}}$
• ${\displaystyle \sec (C)=\frac{\text{hypotenuse}}{\text{adjacent}}}$$\displaystyle \sec{(C)}=\frac{\text{hypotenuse}}{\text{adjacent}}$
• ${\displaystyle \tan (C)=\frac{\text{opposite}}{\text{adjacent}}}$$\displaystyle \tan{(C)}=\frac{\text{opposite}}{\text{adjacent}}$
• ${\displaystyle \sin (C)=\frac{\text{opposite}}{\text{hypotenuse}}}$$\displaystyle \sin{(C)}=\frac{\text{opposite}}{\text{hypotenuse}}$
• ${\displaystyle \csc (C)=\frac{\text{hypotenuse}}{\text{opposite}}}$$\displaystyle \csc{(C)}=\frac{\text{hypotenuse}}{\text{opposite}}$
• ${\displaystyle \cot (C)=\frac{\text{adjacent}}{\text{opposite}}}$$\displaystyle \cot{(C)}=\frac{\text{adjacent}}{\text{opposite}}$

Properties of Trig Functions

• ${\displaystyle \cos (\theta )}$$\displaystyle \cos{(\theta)}$: http://jjw3.com/r/cos.pdf
• ${\displaystyle \sec (\theta )}$$\displaystyle \sec{(\theta)}$: http://jjw3.com/r/sec.pdf
• ${\displaystyle \tan (\theta )}$$\displaystyle \tan{(\theta)}$: http://jjw3.com/r/tan.pdf
• ${\displaystyle \sin (\theta )}$$\displaystyle \sin{(\theta)}$: http://jjw3.com/r/sin.pdf
• ${\displaystyle \csc (\theta )}$$\displaystyle \csc{(\theta)}$: http://jjw3.com/r/csc.pdf
• ${\displaystyle \cot (\theta )}$$\displaystyle \cot{(\theta)}$: http://jjw3.com/r/cot.pdf

Signs of Trigonometric Functions

• $\cos (x)\ge 0$$\cos{(x)}\geq 0$ on $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$$-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}$, i.e., Quadrants I and IV, $y$$y$-axis, and positive $x$$x$-axis
• $\sec (x)\ge 0$$\sec{(x)}\geq 0$ on $-\frac{\pi }{2}<x<\frac{\pi }{2}$$-\frac{\pi}{2}< x<\frac{\pi}{2}$, i.e., Quadrants I and IV and positive $x$$x$-axis
• $\tan (x)\ge 0$$\tan{(x)}\geq 0$ on $0\le x<\frac{\pi }{2}\cup \pi \le x<\frac{3\pi }{2}$$0\leq x<\frac{\pi}{2}\,\cup\,\pi\leq x<\frac{3\pi}{2}$, i.e., Quadrants I and III and positive $x$$x$-axis

Pythagorean Identities

• ${\sin }^2(x)+{\cos }^2(x)=1$$\sin^2{(x)}+\cos^2{(x)}=1$, more specifically,

  • ${\sin }^2(x)=1-{\cos }^2(x)$$\color{red}\sin^2{(x)}=1-\cos^2{(x)}$, and
  • ${\cos }^2(x)=1-{\sin }^2(x)$$\color{red}\cos^2{(x)}=1-\sin^2{(x)}$

• ${\tan }^2(x)+1={\sec }^2(x)$$\color{red}\tan^2{(x)}+1=\sec^2{(x)}$, additionally, ${\tan }^2(x)={\sec }^2(x)-1$$\color{red}\tan^2{(x)}=\sec^2{(x)}-1$

• $1+{\cot }^2(x)={\csc }^2(x)$$\color{red}1+\cot^2{(x)}=\csc^2{(x)}$, additionally, ${\cot }^2(x)={\csc }^2(x)-1$$\color{red}\cot^2{(x)}=\csc^2{(x)}-1$

Evaluation of Trig Functions

See PRE_04i_Trig_Integrals.html.

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Created: Monday, 6 September 2020 13:33 EDT Last Modified: Monday, 10 January 2022 - 07:21 (EST)