Domain Restrictions for Inverse Trig Functions

In this lesson:

• Explain why sine, cosine, and tangent need restricted domains to have inverses
• Identify standard restrictions and define arcsin, arccos, arctan
• Evaluate inverse trig functions at standard values

Learning Objectives for This Lesson

By the end, you will be able to:

1. Explain why trig functions need domain restrictions for inverses
2. State the standard restrictions for sin, cos, and tan
3. Identify the domain and range of each inverse trig function
4. Evaluate inverse trig at standard values

How Many Solutions Does Have?

You know two: and

But there are infinitely many: and for every integer

If "arcsin" means "the angle whose sine is ," which answer should it give?

This ambiguity is why we need domain restriction.

The Horizontal Line Test Fails for Sine

Every horizontal line between and crosses the sine curve infinitely many times.

Three Solutions to

Can you list three solutions?

Why does this show that cosine has no inverse on its full domain?

Each horizontal line cuts the cosine graph infinitely many times — same problem as sine.

Restricting Sine to

On , sine is strictly increasing from to .

• Every output in is achieved exactly once
• The horizontal line test is satisfied
• The inverse is well-defined on this piece

Define: = the unique such that

Arcsin: Restricted Graph and Inverse

• Arcsin domain:
• Arcsin range:
• The range is angles — always between -90° and 90°

Arcsin Domain, Range, and Evaluations

• Domain: ; Range:

Output is always an angle in .

, Not

Both and satisfy .

But the restriction excludes .

The restriction doesn't make wrong — it just makes it outside the domain of arcsin.

Quick Check: Why and Not ?

Explain in your own words:

Why does rather than ?

Your explanation should reference the restricted domain of sine.

Think: what interval is the arcsin output always in?

Restricting Cosine to

On , cosine decreases from to — strictly monotone.

• Full range covered; horizontal line test satisfied

Define: = unique with

• Domain: ; Range:

— not or

Restricting Tangent to

On , tangent increases from to — one-to-one.

Define: = unique with

• Domain: ; Range:

— not

All Three Inverse Functions: Summary Table

The range of each inverse function is different — always check.

Mixed Evaluations: All Three Functions

For each: identify the range, then find the angle in that range.

Quick Check: Domain of Arctan

Is defined? Is defined?

Think: what are the input restrictions for each function?

Could We Have Chosen a Different Restriction?

What if we restricted sine to instead?

• On , sine is strictly decreasing from to
• It's also one-to-one — the horizontal line test is satisfied
• This would give: (not )

Both restrictions are mathematically valid — just different conventions.

Why Use the Standard Domain Restrictions?

Standard restrictions are chosen by convention, not necessity.

Criteria:

• Includes — a natural reference point
• Connected and monotone
• Covers the full range

Any monotone interval covering the full range gives a valid inverse.

Practice: Evaluate and Reason About Inverses

5. Is defined? Why or why not?
6. Propose a different restriction for cosine. What would equal?

Answers to Evaluation Practice Problems

1. ✓ — in
2. ✓ — in
3. ✓ — in
4. , so arcsin gives (not ) ✓
5. Undefined — , outside domain
6. Restrict to →

Key Takeaways from This Lesson

• Periodic functions fail HLT → restrict domain to get inverse
• sin: ; cos: ; tan:
• Arccos and arcsin have different ranges
• only when is in the range
• means inverse function, not reciprocal

What's Next: Solving Trig Equations

HSF.TF.B.7 — Solving Trig Equations

• With arcsin, arccos, arctan defined, you can solve
• Inverse trig gives one solution in the principal range
• The general solution uses periodicity for remaining angles

Today's restrictions make that first step possible.