Inverse Trigonometric Functions

trigonometricfunctions year12 ext1

Inverse trigonometric functions | NSW Curriculum Website

• ME1-12-03 solves problems involving inverse trigonometric functions

📖 Prior Knowledge

Content	Prerequisite relationships
Working with Functions	- Test for even and odd functions → Parity of inverse trig functions - Form and evaluate composite functions → Composites of inverse trig functions
Trigonometry and Measure of Angles	- Find exact ratios for multiples of $\frac{\pi }{6}$ and $\frac{\pi }{4}$ → Evaluate inverse trig expressions - Graph trig functions in radians → Restrict trig functions for inverses
Graph Transformations	- Apply combined transformations → Transform inverse trig graphs
Further Work with Functions	- Identify one-to-one functions → Restrict trig functions for inverses - Find the inverse of a function → Define inverse trigonometric functions - Graph an inverse function → Graph inverse trigonometric functions
Further Trigonometry	- Double angle identities → Double and compound angles with inverse trig

Definitions of inverse trigonometric functions

• Graph $y=\sin x$, $y=\cos x$ and $y=\tan x$, for $-2\pi \le x\le 2\pi$ using graphing applications and recognise that these three functions fail the horizontal line test
• Examine possible domain restrictions of $y=\sin x$, $y=\cos x$ and $y=\tan x$ to obtain the corresponding inverse functions using graphing applications
• Define ${\sin }^{-1}x$ or $\arcsin x$ to be the inverse function of $y=\sin x$ restricted to $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$, and determine the domain and range of $y={\sin }^{-1}x$
• Define ${\cos }^{-1}x$ or $\arccos x$ to be the inverse function of $y=\cos x$ restricted to $0\le x\le \pi$, and determine the domain and range of $y={\cos }^{-1}x$
• Define ${\tan }^{-1}x$ or $\arctan x$ to be the inverse function of $y=\tan x$ restricted to $-\frac{\pi }{2}<x<\frac{\pi }{2}$, and determine the domain and range of $y={\tan }^{-1}x$
• Evaluate and simplify expressions, and prove results using the definitions of ${\sin }^{-1}x$, ${\cos }^{-1}x$ and ${\tan }^{-1}x$

Graphs of inverse trigonometric functions

• Graph $y=\sin x$ for $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$, then use a reflection in $y=x$ to graph $y={\sin }^{-1}x$ with and without graphing applications
• Graph $y=\cos x$ for $0\le x\le \pi$, then use a reflection in $y=x$ to graph $y={\cos }^{-1}x$ with and without graphing applications
• Graph $y=\tan x$ for $-\frac{\pi }{2}<x<\frac{\pi }{2}$, then use a reflection in $y=x$ to graph $y={\tan }^{-1}x$ with and without graphing applications
• Classify $y={\sin }^{-1}x$, $y={\cos }^{-1}x$ and $y={\tan }^{-1}x$ as odd, even, or neither odd nor even
• Examine the properties ${\sin }^{-1}(-x)=-{\sin }^{-1}x$, ${\cos }^{-1}(-x)=\pi -{\cos }^{-1}x$, ${\tan }^{-1}(-x)=-{\tan }^{-1}x$ and ${\cos }^{-1}x+{\sin }^{-1}x=\frac{\pi }{2}$ using graphing applications and prove them algebraically
• Use the properties ${\sin }^{-1}(-x)=-{\sin }^{-1}x$, ${\cos }^{-1}(-x)=\pi -{\cos }^{-1}x$, ${\tan }^{-1}(-x)=-{\tan }^{-1}x$ and ${\cos }^{-1}x+{\sin }^{-1}x=\frac{\pi }{2}$ to solve problems, simplify expressions and prove results
• Graph the composite functions $y=\sin ({\sin }^{-1}x)$, $y=\cos ({\cos }^{-1}x)$ and $y=\tan ({\tan }^{-1}x)$ by considering their domains and ranges
• Examine the graphs of the composite functions $y=\sin ({\sin }^{-1}x)$, $y=\cos ({\cos }^{-1}x)$ and $y=\tan ({\tan }^{-1}x)$, establish the values of $x$ for which the formulas $\sin ({\sin }^{-1}x)=x$, $\cos ({\cos }^{-1}x)=x$ and $\tan ({\tan }^{-1}x)=x$ are true, and apply the formulas to solve problems, simplify expressions and prove results
• Graph the composite functions $y={\sin }^{-1}(\sin x)$, $y={\cos }^{-1}(\cos x)$ and $y={\tan }^{-1}(\tan x)$ by considering their domains and ranges
• Examine the graphs of the composite functions $y={\sin }^{-1}(\sin x)$, $y={\cos }^{-1}(\cos x)$ and $y={\tan }^{-1}(\tan x)$, establish the values of $x$ for which the formulas ${\sin }^{-1}(\sin x)=x$, ${\cos }^{-1}(\cos x)=x$ and ${\tan }^{-1}(\tan x)=x$ are true, and apply the formulas to solve problems, simplify expressions and prove results
• Use graphing applications to explore reflections, translations and dilations of the functions $f(x)={\sin }^{-1}x$, $f(x)={\cos }^{-1}x$ and $f(x)={\tan }^{-1}x$, and confirm that the principles of transformations hold for the inverse trigonometric functions
• Apply the principles of transformations to inverse trigonometric functions to determine the function rule and graph the function, finding the domain and range of the transformed graph and determining any intercepts and asymptotes where appropriate