Sin, Cos, Tan Inverse Functions- Usage and Examples

What Are Inverse Trigonometric Functions?

Inverse trig functions let you work backwards. Regular trig functions take an angle and give you a ratio. Inverse trig functions take a ratio and give you an angle.

Here's the simple version:

• sin(30°) = 0.5 → angle gives you ratio
• arcsin(0.5) = 30° → ratio gives you angle

The three main inverse functions are:

• arcsin(x) or sin⁻¹(x) — inverse sine
• arccos(x) or cos⁻¹(x) — inverse cosine
• arctan(x) or tan⁻¹(x) — inverse tangent

The notation sin⁻¹(x) trips people up. It's not 1/sin(x). That's csc(x). The ⁻¹ means "inverse function," not "reciprocal."

The Domain and Range Problem

Inverse trig functions don't accept just any input. They have restrictions because trig functions aren't one-to-one — they repeat values.

For example, sin(30°) = 0.5 but sin(150°) = 0.5 too. The inverse function can only return one answer. That's why calculators restrict the range.

Standard Ranges

• arcsin(x): returns angles from -90° to 90° (or -π/2 to π/2 in radians)
• arccos(x): returns angles from 0° to 180° (or 0 to π radians)
• arctan(x): returns angles from -90° to 90° (or -π/2 to π/2 radians)

The input (domain) for arcsin and arccos is always -1 to 1. Anything outside that range is impossible — there's no angle whose sine or cosine exceeds those bounds.

Quick Reference Table

How to Calculate Inverse Trig Functions

Using a Calculator

Most scientific calculators have these functions built-in. Look for buttons labeled:

• sin⁻¹ or ASIN
• cos⁻¹ or ACOS
• tan⁻¹ or ATAN

Make sure your calculator is in the right mode. Degrees vs radians matters. If you're working in degrees, check that the calculator is set to DEG, not RAD.

Common Values to Memorize

These come up constantly in problems and tests:

Solved Examples

Example 1: Basic arcsin

Find arcsin(0.866)

Using a calculator: arcsin(0.866) ≈ 60°

You can verify this: sin(60°) = 0.866 ✓

Example 2: arccos in a real problem

A right triangle has an adjacent side of 5 and hypotenuse of 10. Find the angle.

cos(θ) = adjacent/hypotenuse = 5/10 = 0.5

θ = arccos(0.5) = 60°

Example 3: arctan for slope

A roof rises 4 feet vertically over a horizontal distance of 12 feet. What is the angle of elevation?

tan(θ) = rise/run = 4/12 = 0.333

θ = arctan(0.333) ≈ 18.4°

This is where arctan shines. It accepts any real number as input, making it perfect for slope problems.

Example 4: Finding the angle in a right triangle

Opposite side = 3, Adjacent side = 4. Find the angle.

First, find the tangent: tan(θ) = opposite/adjacent = 3/4 = 0.75

Then invert: θ = arctan(0.75) ≈ 36.9°

Practical Applications

Inverse trig functions show up everywhere in real work:

• Engineering — calculating forces and angles in structures
• Physics — projectile motion, vector analysis
• Computer graphics — rotations, camera angles
• Navigation — bearing calculations
• Surveying — determining heights and distances

Getting Started: Your Quick Checklist

1. Identify what you know — a ratio (sine, cosine, or tangent value)
2. Choose the right function — match the ratio to sin, cos, or tan
3. Check your input — arcsin and arccos need values between -1 and 1
4. Set your mode — degrees or radians, pick one and stick with it
5. Calculate — use the inverse function to get your angle
6. Verify — plug the angle back into the original trig function to check

Common Mistakes to Avoid

• Confusing sin⁻¹(x) with csc(x) — they're not the same
• Using degrees when the problem expects radians — or vice versa
• Trying to take arcsin of a number greater than 1 — it won't work
• Forgetting that arccos(0.5) = 60° but arcsin(0.5) = 30° — same input, different output

The Relationship Between the Three Functions

There's a useful identity:

arccos(x) = π/2 - arcsin(x)

Or in degrees: arccos(x) = 90° - arcsin(x)

This means if you know arcsin, you automatically know arccos. For example:

• arcsin(0.5) = 30°
• arccos(0.5) = 90° - 30° = 60°

Arctan doesn't have this simple relationship with the other two, but it connects through the identity:

arctan(x) = arcsin(x / √(1 + x²))

This comes in handy when you need to compute arctan on a calculator that only has arcsin.

When to Use Which Function

Pick your function based on which ratio you have:

• Use arcsin when you have opposite/hypotenuse
• Use arccos when you have adjacent/hypotenuse
• Use arctan when you have opposite/adjacent

If you're not sure, draw the triangle. The side positions tell you which trig ratio applies.

That's it. Inverse trig functions are straightforward once you understand the work-backwards logic. Practice with a few triangles, and it'll click.