Inverse Trigonometric Ratios - Definition & Examples

TL;DR

Inverse trigonometric ratios run the ordinary ratios backwards: you give them a ratio of sides and they return the angle that produced it. Written sin⁡−1, cos⁡−1, tan⁡−1 (or arcsin, arccos, arctan), they answer "what angle has this sine?" This article defines all six, gives their domain and range, untangles the inverse-versus-reciprocal trap, and works six examples.

What Are Inverse Trigonometric Ratios?

Inverse trigonometric ratios are the operations that recover an angle from a known trigonometric ratio. If sin⁡θ=12, then the inverse sine undoes the sine to give θ: sin⁡−1(12)=30°. There are six, one for each ordinary ratio:

• sin⁡−1x (arcsin⁡x)
• cos⁡−1x (arccos⁡x)
• tan⁡−1x (arctan⁡x)
• csc⁡−1x
• sec⁡−1x
• cot⁡−1x

The "arc" name comes from the unit circle: the inverse hands back the arc (the angle) that wraps to a given coordinate. One warning belongs right at the top, before any example uses the notation: the −1 in sin⁡−1x marks an inverse function, not an exponent. It does not mean 1/sin⁡x.

Domain And Range Of The Inverse Trigonometric Ratios

Because the ordinary ratios repeat, many angles share the same value — so each inverse is restricted to one stretch of angles where it is single-valued. That restricted output stretch is its range (also called the principal-value branch); the allowed inputs form its domain.

Inverse Versus Reciprocal — The Trap That Costs The Most Marks

Is sin⁡−1x the same as 1/sin⁡x? No — confusing them is the single biggest error in this topic. They are different operations that happen to share a misleading notation:

• Inverse: sin⁡−1x (arcsin) takes a ratio and returns an angle. For example, sin⁡−1(12)=π/6.
• Reciprocal: 1/sin⁡x is the cosecant, csc⁡x — it takes an angle and returns a flipped ratio.

The −1 superscript means "inverse function" here, the same way f−1(x) means the inverse of f. It is not the exponent −1.

Examples Of Inverse Trigonometric Ratios

Example 1

Find sin⁡−1(12).

Ask: what angle in the range [−π/2, π/2] has a sine of 1/2?

sin⁡π/6=1/2⇒sin⁡−1(1/2)=π/6.

Final answer: π/6.

Example 2

A right triangle has an opposite side of 3 and a hypotenuse of 6. Find the angle.

sin⁡θ=opposite/hypotenuse, so we can express it as sin⁡θ=3/6=1/2.

θ=sin⁡−1(1/2)=π/6=30°.

Final answer: 30°.

Example 3

Find cos⁡−1(√3/2).

What angle in [0, π] has cosine √3/2?

cos⁡π/6=√3/2⇒cos⁡−1(√3/2)=π/6.

Final answer: π/6.

Example 4

A ramp rises 5 m over a horizontal run of 5 m. What angle does it make with the ground?

Using the inverse tangent:

tan⁡θ=5/5=1.

θ=tan⁡−1(1)=π/4=45°.

Final answer: 45°.

Example 5

Find tan⁡−1(3).

What angle in (−π/2, π/2) has tangent 3?

tan⁡π/3=3⇒tan⁡−1(3)=π/3.

Final answer: π/3.

Example 6

Evaluate cos⁡−1(−1/2).

The input is negative, so the answer must be an obtuse angle in Quadrant II. The reference angle is π/3.

cos⁡−1(−1/2)=π−π/3=2π/3.

Final answer: 2π/3.

Why The Inverse Ratios Exist

Trigonometry was built to find unreachable lengths — but turn the problem around and you often have the lengths and need the angle instead. The inverse ratios exist because "what angle?" is just as real a question as "what length?"

• Applications include:
  • Surveying and construction.
  • Navigation and robotics.
  • Calculus.

Tripping Points in Inverse Trigonometric Ratios to Avoid

Mistake 1

Treating sin⁡−1x as 1/sin⁡x.

Correct way: sin⁡−1 returns an angle; 1/sin⁡x returns a flipped ratio.

Mistake 2

Asking for an inverse of an input outside the domain.

Correct way: Trust that inputs for sin⁡−1 and cos⁡−1 cannot exceed ±1.

Mistake 3

Forgetting the range and returning the wrong angle.

Correct way: Always check the range of values for inverse ratios.

Key Takeaways

• Inverse trigonometric ratios take a ratio and return the angle that produced it.
• The six are sin⁡−1, cos⁡−1, tan⁡−1, csc⁡−1, sec⁡−1, cot⁡−1.
• The −1 means inverse, not reciprocal.
• Always return angles within the function's range.