Unit 6: Inverse Trigonometric Functions
Restricting sine, cosine, and tangent so they pass the horizontal line test, and using their inverses to solve equations and simplify compositions.
5 topics · 18 questions
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Restricting Domains to Define Inverses
- How it worksHow does restricting sine to the interval from -π/2 to π/2 make it pass the horizontal line test?
- Common mistakeWhy isn't the restricted domain of sine an arbitrary choice, and what makes -π/2 to π/2 the right one?
- Going furtherWhat happens to the restricted domain and its inverse when you do the same process for cosine instead of sine?
Your turn1 of 2
A student says the choice of [-π/2, π/2] for restricting sine is just an arbitrary convention that could have just as easily been any other interval where sine is one-to-one. What's the flaw?The Inverse Sine Function
- How it worksHow does arcsin(x) reverse sine by only ever returning an angle between -π/2 and π/2?
- Worked exampleFind the exact value of arcsin(√2/2).
- Common mistakeWhy isn't arcsin(x) the same thing as 1/sin(x)?
- Common mistakeWhy doesn't arcsin(2) exist as a real angle, even though sine itself is defined for every angle?
Your turn1 of 2
Let f(x) = sin(x) with domain restricted to [-π/2, π/2]. Write f−1(x).The Inverse Cosine and Inverse Tangent Functions
- How it worksHow do the restricted ranges of arccos and arctan differ from the restricted range of arcsin?
- Worked exampleFind the exact values of arccos(-1/2) and arctan(1).
- Common mistakeWhy doesn't arctan(x) ever output an angle of exactly π/2 or -π/2, no matter how large x gets?
- Ties togetherHow does arctan(x) approaching π/2 as x grows without bound preview the idea of a limit at infinity?
Your turn1 of 2
Let f(x) = tan(x) with domain restricted to (-π/2, π/2). Write f−1(x).Composing Trig and Inverse Trig Functions
- How it worksHow does drawing a right triangle from an inverse trig expression let you simplify a composition like cos(arcsin(x))?
- Worked exampleFind the exact value of sin(arccos(3/5)) using a right triangle.
- Common mistakeWhy doesn't arcsin(sin(x)) always just simplify back to x?
Your turn1 of 2
Write sin(arccos(2·x)) as an algebraic expression in x, with no trig functions.Solving Equations with Inverse Trig Functions
- How it worksHow do you use arcsin together with the unit circle's symmetry to find every solution in one cycle?
- Worked exampleFind all solutions to sin(x) = 0.5 on the interval from 0 to 2pi.
- Worked exampleSolve 2·cos(x) - 1 = 0 for every x between 0 and 2pi.
- Common mistakeWhy doesn't a calculator's arcsin button hand you every solution to a trig equation at once?
Your turn1 of 2
Find the solution to sin(x) = 0.692 in the interval [0, π/2].