Inverses of trigonometric functions

Learning Objectives Understand and explain why trigonometric functions must have their domains restricted to define an inverse function. Define and graph the inverse sine, cosine, and tangent functions, stating their respective domains and ranges. Evaluate inverse trigonometric expressions for exact values, both with and without a calculator. Simplify composite expressions involving trigonometric and inverse trigonometric functions, such as sin(arccos(x)). Solve trigonometric equations for a principal value using inverse functions. Calculate the derivatives of the inverse trigonometric functions, applying the chain rule where necessary. If you know a skyscraper is 400m tall and you're standing 300m away, how can you find the exact angle to tilt your head to see the top?...

TermDefinitionExample Inverse FunctionA function that reverses the action of another function. If f(a) = b, then the inverse function, denoted f⁻¹(x), satisfies f⁻¹(b) = a.If f(x) = 2x, then its inverse is f⁻¹(x) = x/2. For instance, f(3) = 6 and f⁻¹(6) = 3. One-to-One FunctionA function where every distinct input produces a distinct output. A function must be one-to-one to have an inverse. Graphically, it must pass the Horizontal Line Test.y = x³ is one-to-one. y = x² is not, because f(2) = 4 and f(-2) = 4. Domain RestrictionThe process of limiting the domain of a function that is not one-to-one to create a new, well-behaved function that *is* one-to-one, thus allowing an inverse to be defined.The function f(x) = sin(x) is not one-to-one. We restrict its domain to [-π/2, π/2] to define i...

Inverse Composition Rules 1. sin(arcsin(x)) = x for -1 ≤ x ≤ 1 \\ 2. arcsin(sin(y)) = y for -π/2 ≤ y ≤ π/2 \\ 3. cos(arccos(x)) = x for -1 ≤ x ≤ 1 \\ 4. arccos(cos(y)) = y for 0 ≤ y ≤ π \\ 5. tan(arctan(x)) = x for all real x \\ 6. arctan(tan(y)) = y for -π/2 < y < π/2 These rules describe how a trigonometric function and its inverse 'cancel' each other out. It is critical to check that the variable (x or y) is within the specified domain or range for the rule to apply directly. Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x. \\ 1. \frac{d}{dx}(\arcsin(u)) = \frac{u'}{\sqrt{1-u^2}} \\ 2. \frac{d}{dx}(\arccos(u)) = -\frac{u'}{\sqrt{1-u^2}} \\ 3. \frac{d}{dx}(\arctan(u)) = \frac{u'}{1+u^2} These are fundame...