The Inverse Function Restrictor
Inverse trig functions, solving triangles, and compositions
Step 1 of 8: The Same Old Problem
Remember \(x^2\) from Simulation 3? Squaring gives the same output for different inputs: \((-3)^2 = 9\) and \(3^2 = 9\). Sine does the same thing. Move the slider and watch how many x-values share the same sine value.
Try at least 3 different y-values to continue.
Step 2 of 8: Reflecting Over \(y = x\)
To find the inverse, we reflect the graph over \(y = x\). Press the button to reflect the sine curve and see what happens.
Step 3 of 8: Restrict the Domain
Fix it the way we fixed \(x^2\): restrict the domain. Use the sliders to narrow which part of the sine curve is visible. The restriction needs to do two jobs: pass the horizontal line test (one-to-one) and cover the full range \([-1, 1]\). Experiment freely.
Horizontal line test (one-to-one):
Covers full range [-1, 1]:
Step 4 of 8: Why \([0, \pi]\) Fails
Try the interval \([0, \pi]\). It passes the horizontal line test. But look at its range. What values can sine actually reach on \([0, \pi]\)?
Step 5 of 8: Why \([0, 2\pi]\) Fails
Now try \([0, 2\pi]\). It covers the full range \([-1, 1]\). Does it pass the horizontal line test?
Step 6 of 8: The Right Window
Go back to the adjustable window. Find the shortest interval that does BOTH jobs: one-to-one and covers \([-1, 1]\). Both indicators must show green.
Horizontal line test (one-to-one):
Covers full range [-1, 1]:
Step 7 of 8: Cosine and Tangent
Now predict the restricted domains for cosine and tangent. For each, think: which interval is one-to-one and covers the full range?
Cosine: What is the standard restricted domain for \(\arccos\)?
Step 8 of 8: Compositions and Surprises
Three compositions. Each one behaves differently. Work through them in order.
Case 1: \(\sin(\arcsin(0.7))\)
What should this equal?
Explore Inverse Trig Functions
Horizontal line test (one-to-one):
Covers full range:
Select a function and adjust the domain window. The tests update in real time.
Composition Calculator
Pick a composition and see the step-by-step trace.
Apply What You Learned
Starter: Evaluate Inverse Trig Values
Evaluate \(\arcsin\!\left(\frac{\sqrt{3}}{2}\right)\). In which interval does the answer live, and why?
Evaluate \(\arccos\!\left(-\frac{1}{2}\right)\). In which interval does the answer live, and why?
Stretch: Triangle Without a Calculator
Evaluate \(\sin(\arccos(\frac{5}{13}))\) without a calculator.
\(\arccos(\frac{5}{13})\) gives you an angle. Draw the triangle: adjacent = 5, hypotenuse = 13. Find the opposite side, then find sine.
What is the opposite side?
Challenge: The Ramp Problem
A ramp gains 2 feet of elevation for every 5 feet of ramp length.
Express the ramp angle using an inverse trig function.
Select a mode above to begin exploring inverse trig functions.
- Explain why the full sine, cosine, and tangent functions do not have inverses
- Identify the standard restricted domains for arcsin, arccos, and arctan and justify why each is chosen
- Evaluate inverse trig expressions by reasoning about the restricted domain
- Compute compositions like \(\sin(\arccos(x))\) by building right triangles, without a calculator
- Recognize that \(\arcsin(\sin(x)) = x\) only when x is in the restricted domain
Quick Check
What is \(\arcsin\!\left(\sin\!\left(\frac{7\pi}{6}\right)\right)\)?
Try This
Starter
Evaluate \(\arcsin(\frac{\sqrt{3}}{2})\) and \(\arccos(-\frac{1}{2})\). For each, explain in one sentence why the answer lives where it does, using the restricted domain.
Stretch
Evaluate \(\sin(\arccos(\frac{5}{13}))\) without a calculator. Draw the triangle. Which quadrant does \(\arccos(\frac{5}{13})\) land in, and how do you know?
Challenge
A ramp gains 2 feet of elevation for every 5 feet of ramp length. Express the angle using an inverse trig function. Then find the cosine of that angle without ever computing the angle. Then check: does your answer match \(\cos(\arcsin(\frac{2}{5}))\)?
Instructor Notes
Teaching Notes
This simulation builds the restricted domain from scratch rather than presenting it as a definition. Students discover that \([-\pi/2, \pi/2]\) is not arbitrary by watching two other intervals fail. The "why that one?" question from the hook resolves through experiment, not lecture.
Step 8 is the payoff: the three composition cases reveal that restricting the domain has consequences students do not expect. Case 2 (\(\arcsin(\sin(5\pi/4)) = -\pi/4\)) is a designed-failure step. Most students predict \(5\pi/4\) and the surprise is the teaching moment. Do not soften it or hint the answer.
Common Student Errors
- Believing \(\arcsin(\sin(x)) = x\) for every x. This is the central misconception and Step 8 Case 2 targets it directly.
- Confusing \(\arcsin(x)\) with \(\frac{1}{\sin(x)}\). Reinforce that arcsin is the inverse function, not the reciprocal. \(\frac{1}{\sin(x)}\) is cosecant.
- Choosing \([0, \pi]\) for sine because it "looks right" without checking whether negative outputs are covered.
- Using \([-\pi/2, \pi/2]\) for all three inverse functions. Cosine uses \([0, \pi]\) and tangent uses the open interval \((-\pi/2, \pi/2)\).
Discussion Questions
- Why does your calculator give only one answer for arcsin(0.5)? Does that mean the other answers do not exist?
- Could we choose a different restricted domain for sine, say \([\pi/2, 3\pi/2]\)? Would it work? Would there be a reason to prefer it?
- In the triangle method for compositions, why do we never need to compute the angle at all?
Exam Connection
Evaluating inverse trig compositions (like \(\sin(\arccos(x))\) or \(\arcsin(\sin(\theta))\) outside the restricted domain) is one of the most common exam topics in this unit. The triangle method from Step 8 Case 3 is the standard approach students need. Step 8 Case 2 tests the restricted-domain concept directly, and this is where most exam errors come from.