Inverse Trigonometric Functions Calculator

Ever stared at a triangle and thought, "I know these two sides, but what's the angle?" That's exactly what this calculator solves.

Inverse trigonometric functions work backwards from what you learned in trig class. Instead of plugging in an angle to get a ratio, you plug in a ratio and get the angle back. This calculator handles all six inverse functions—arcsin, arccos, arctan, arccot, arcsec, and arccsc—and shows your result in both degrees and radians so you don't have to convert anything yourself.

Whether you're powering through a precalculus assignment at midnight, figuring out the slope angle for a construction project, or writing game code that needs rotation values, just select your function, enter your number, and you've got your answer.

Let's break this down simply.

Regular trig functions are like a one-way street: you put in an angle, and out comes a ratio. Sin(30°) gives you 0.5. Easy enough.

But what if you're going the other direction? What if you have 0.5 and need to find the angle? That's where inverse trig functions come in. Arcsin(0.5) = 30°. You're essentially asking the function, "Hey, which angle has a sine of 0.5?"

A quick note on notation: You'll sometimes see these written as sin⁻¹, cos⁻¹, and tan⁻¹. Don't let that confuse you—the superscript -1 doesn't mean "divide by sine." It's just math's way of saying "inverse." The "arc" names (arcsin, arccos, arctan) mean the exact same thing and tend to cause less confusion, which is why this calculator uses them.

Here's when you'd use each one:

If you're just starting out, focus on the big three: arcsin, arccos, and arctan. They handle about 95% of what most people need.

Here's something that trips people up: not every number works for every function. Try to find arcsin(2) and you'll get an error—not because the calculator is broken, but because no angle exists whose sine equals 2. (Sine values max out at 1.)

Here's the full breakdown:

Why the restricted ranges? These are called "principal values," and they exist because technically, infinitely many angles could give you the same sine or cosine. (Sin(30°) and sin(150°) both equal 0.5, for instance.) The calculator picks one definitive answer from a sensible range so you get a consistent result every time.

1. Pick your function from the dropdown. Not sure which one? Here's the quick test:

• Do you have opposite and hypotenuse? → arcsin
• Do you have adjacent and hypotenuse? → arccos
• Do you have opposite and adjacent (like rise and run)? → arctan

2. Type in your value. This is usually a ratio you've calculated—like 0.5, or 0.866, or 1.5 for arctan.

3. Read both results. You'll see the angle in degrees and radians. Grab whichever one you need.

That's it. No mode switching, no conversion steps, no guesswork.

You're given a right triangle where the opposite side is 7 and the hypotenuse is 14. What's the angle?

Step 1: Calculate your ratio: 7 ÷ 14 = 0.5 Step 2: Select arcsin, enter 0.5 Result: 30° (or π/6 radians, which is about 0.5236)

If your teacher wants the answer in radians, you've got it. Degrees? Also covered.

You're building a wheelchair ramp. It rises 1 foot over a horizontal distance of 12 feet. Is this within the ADA's maximum slope requirement of about 4.8 degrees?

Step 1: Calculate rise over run: 1 ÷ 12 = 0.0833 Step 2: Select arctan, enter 0.0833 Result: 4.76°

Good news—you're just under the limit.

You're coding an enemy AI that needs to face the player. The player is at coordinates (8, 6) relative to the enemy at the origin. What rotation angle should the enemy sprite use?

Step 1: Calculate opposite over adjacent: 6 ÷ 8 = 0.75 Step 2: Select arctan, enter 0.75 Result: 36.87° (or 0.6435 radians)

Most game engines use radians internally, so you'd probably grab that 0.6435 value.

In a calculus problem, you're told sec(θ) = 2 and need to find θ.

Step 1: Select arcsec, enter 2 Result: 60° (or π/3 radians)

You can verify: cos(60°) = 0.5, and sec = 1/cos, so sec(60°) = 1/0.5 = 2. ✓

Both give you the same angle—it's like measuring distance in miles versus kilometers. Here's when to use each:

Go with degrees when:

• You're doing geometry homework or taking a standardized test
• You're explaining something to someone who isn't a math person
• You're working on anything physical: construction, navigation, DIY projects
• The problem specifically asks for degrees (obviously)

Go with radians when:

• You're in calculus (derivatives and integrals of trig functions need radians)
• You're programming (JavaScript, Python, C++, and most languages default to radians)
• You're doing physics with angular velocity, circular motion, or waves
• You're working with the unit circle

Quick conversion if you ever need it:

• Degrees → Radians: multiply by π/180
• Radians → Degrees: multiply by 180/π

Or just... use this calculator and get both.

Construction and Architecture Calculating roof pitch, stair angles, ramp slopes. An architect saying "the roof has a 6:12 pitch" is giving you a tangent ratio—arctan converts that to the actual angle.

Programming and Game Development Any time you convert between (x, y) coordinates and angle/distance, you're using inverse trig. Character movement, projectile trajectories, camera angles, procedural generation—it comes up constantly.

Navigation GPS systems, flight planning, sailing. Finding the bearing between two points on a map involves inverse tangent calculations.

Physics Projectile motion (finding launch angles), optics (angles of reflection and refraction), mechanics (force components). Physics and inverse trig are inseparable.

Surveying and Engineering Measuring angles in the field, calculating load distributions, designing mechanical linkages. If there's a triangle involved, there's probably an inverse trig function nearby.

Entering impossible values If you're trying arcsin(1.5) and getting an error, that's the math telling you something—no angle has a sine of 1.5. Double-check your ratio calculation.

Thinking sin⁻¹ means 1/sin This catches a lot of people. Sin⁻¹(x) is the inverse sine function, not 1 divided by sin(x). If you want the reciprocal of sine, that's cosecant.

Using the wrong function entirely Before you calculate, label your triangle sides: which is opposite the angle you want, which is adjacent, and which is the hypotenuse? Then match to the right function.

Getting a negative angle and panicking Totally normal. Arcsin and arctan return negative angles for negative inputs. A negative angle just means it's measured clockwise instead of counterclockwise. It's mathematically correct.

Mixing up your units at the end You calculated in radians but your teacher wanted degrees (or vice versa). Always double-check what format is expected before writing your final answer.

If you want the precise mathematical definitions:

• arcsin(x) returns the angle y where sin(y) = x, with y between -π/2 and π/2
• arccos(x) returns the angle y where cos(y) = x, with y between 0 and π
• arctan(x) returns the angle y where tan(y) = x, with y between -π/2 and π/2
• arccot(x) returns the angle y where cot(y) = x, with y between 0 and π
• arcsec(x) returns the angle y where sec(y) = x, with y between 0 and π (excluding π/2)
• arccsc(x) returns the angle y where csc(y) = x, with y between -π/2 and π/2 (excluding 0)

The restricted ranges guarantee exactly one answer for any valid input—no ambiguity.

Struggling with which function to use? Start with the big three—arcsin, arccos, and arctan—and you'll handle most problems you encounter. The others are there when you need them.