You've mastered using sine, cosine, and tangent to find sides of triangles when you know the angles. But what about the reverse problem? What if you know the sides and need to find the angles? This is where inverse trigonometric functions come in—mathematical tools that work backwards from ratios to angles.
Understanding inverse trig functions transforms you from someone who can follow instructions to someone who can solve complete triangle problems. Let's explore how these functions work, why they're necessary, and how to use them effectively.
The Fundamental Problem
Consider this scenario:
Forward problem (using regular trig):
- Given: angle = 30°, hypotenuse = 10
- Find: opposite side
- Solution: sin(30°) = opposite/10 → opposite = 10 × 0.5 = 5
Reverse problem (needs inverse trig):
- Given: opposite = 5, hypotenuse = 10
- Find: the angle
- Question: What angle has sin(angle) = 5/10 = 0.5?
This reverse question requires inverse trigonometric functions.
What "Inverse" Means
An inverse function undoes what the original function does.
Forward: angle → ratio
sin(30°) = 0.5 (the angle 30° produces the ratio 0.5)
Inverse: ratio → angle
sin⁻¹(0.5) = 30° (the ratio 0.5 came from the angle 30°)
The notation:
- sin⁻¹ is called "inverse sine" or "arcsine"
- cos⁻¹ is called "inverse cosine" or "arccosine"
- tan⁻¹ is called "inverse tangent" or "arctangent"
Important: The "⁻¹" in sin⁻¹ doesn't mean "to the negative one power" (that would be 1/sin). It means "inverse function." This confuses many students initially.
Alternative notation:
- arcsin(x) = sin⁻¹(x)
- arccos(x) = cos⁻¹(x)
- arctan(x) = tan⁻¹(x)
Both notations mean the same thing. Calculators typically use sin⁻¹, while mathematicians often prefer arcsin.
The Three Inverse Functions
Inverse Sine: sin⁻¹ or arcsin
Question it answers: "What angle has this sine value?"
Domain (valid inputs): -1 ≤ x ≤ 1
(Sine values can't exceed 1 or go below -1)
Range (possible outputs): -90° ≤ angle ≤ 90°
(The function returns angles in this range)
Example:
- sin⁻¹(0.5) = 30°
- sin⁻¹(0.707) ≈ 45°
- sin⁻¹(0.866) ≈ 60°
- sin⁻¹(1) = 90°
On your calculator: Look for a button labeled "sin⁻¹" or "arcsin" (often requires pressing a "2nd" or "shift" button first).
Inverse Cosine: cos⁻¹ or arccos
Question it answers: "What angle has this cosine value?"
Domain (valid inputs): -1 ≤ x ≤ 1
(Cosine values can't exceed 1 or go below -1)
Range (possible outputs): 0° ≤ angle ≤ 180°
(The function returns angles in this range)
Example:
- cos⁻¹(0.5) = 60°
- cos⁻¹(0.707) ≈ 45°
- cos⁻¹(0.866) ≈ 30°
- cos⁻¹(0) = 90°
On your calculator: Look for "cos⁻¹" or "arccos" button.
Inverse Tangent: tan⁻¹ or arctan
Question it answers: "What angle has this tangent value?"
Domain (valid inputs): All real numbers
(Tangent can be any value)
Range (possible outputs): -90° < angle < 90°
(The function returns angles in this range, excluding exactly ±90°)
Example:
- tan⁻¹(0) = 0°
- tan⁻¹(0.577) ≈ 30°
- tan⁻¹(1) = 45°
- tan⁻¹(1.732) ≈ 60°
- tan⁻¹(10) ≈ 84.29°
On your calculator: Look for "tan⁻¹" or "arctan" button.
Why the Restricted Ranges?
You might wonder why inverse functions return specific angle ranges. The answer lies in how functions work mathematically.
The problem: Multiple angles can have the same sine, cosine, or tangent value.
Example:
- sin(30°) = 0.5
- sin(150°) = 0.5 (also!)
If sin⁻¹(0.5) could return any angle with sine = 0.5, it would return infinitely many values: 30°, 150°, 390°, 510°, etc.
The solution: Inverse functions return the principal value—the angle in a standard range. When working with angles, whether measured in degrees or radians, these principal values provide consistent answers.
For right triangle problems (our focus), this works perfectly because:
- Right triangle angles are between 0° and 90°
- All three inverse functions return values in or including this range
- You'll always get the right answer for right triangle problems
- The unit circle provides a complete geometric view of why these ranges are chosen
Using Inverse Functions to Find Angles
Step-by-Step Process
Step 1: Identify what sides you know
Step 2: Determine which ratio those sides create
Step 3: Calculate the ratio (as a decimal)
Step 4: Use the appropriate inverse function
Step 5: Verify your answer makes sense
Example 1: Finding an Angle with Sine
Given: In a right triangle, opposite = 7, hypotenuse = 25. Find the angle.
Step 1: We have opposite and hypotenuse
Step 2: This creates the sine ratio
Step 3: sin(angle) = 7/25 = 0.28
Step 4: angle = sin⁻¹(0.28) ≈ 16.26°
Step 5: Check: Is 16.26° reasonable? Yes, it's acute as expected.
Example 2: Finding an Angle with Cosine
Given: In a right triangle, adjacent = 12, hypotenuse = 15. Find the angle.
Step 1: We have adjacent and hypotenuse
Step 2: This creates the cosine ratio
Step 3: cos(angle) = 12/15 = 0.8
Step 4: angle = cos⁻¹(0.8) ≈ 36.87°
Step 5: Check: Is 36.87° reasonable? Yes.
Example 3: Finding an Angle with Tangent
Given: In a right triangle, opposite = 8, adjacent = 15. Find the angle.
Step 1: We have opposite and adjacent (both legs)
Step 2: This creates the tangent ratio
Step 3: tan(angle) = 8/15 ≈ 0.533
Step 4: angle = tan⁻¹(0.533) ≈ 28.07°
Step 5: Check: Is 28.07° reasonable? Yes.
Finding the Other Angle
Remember: in a right triangle, the two acute angles sum to 90°.
Method 1: Subtract from 90°
If you found angle A = 28.07°, then angle B = 90° - 28.07° = 61.93°
Method 2: Use inverse trig on the complementary ratio
For the same triangle, from angle B's perspective:
- What was opposite to A is adjacent to B
- tan(B) = 15/8 = 1.875
- B = tan⁻¹(1.875) ≈ 61.93°
Both methods give the same answer!
Common Mistakes and How to Avoid Them
Mistake 1: Confusing the Function and Its Inverse
Error: Thinking sin⁻¹ means 1/sin
Reality:
- sin⁻¹ is the inverse function (ratio → angle)
- 1/sin is the reciprocal (also called cosecant)
- These are completely different!
Example:
- sin⁻¹(0.5) = 30° (inverse function)
- 1/sin(30°) = 1/0.5 = 2 (reciprocal)
Mistake 2: Calculator in Wrong Mode
Error: Getting weird answers like tan⁻¹(1) = 0.785 instead of 45°
Reality: Your calculator is in radian mode, not degree mode.
Solution:
- Check the mode indicator on your calculator
- 0.785 radians ≈ 45° (it's the right answer in radians)
- Switch to degree mode for most practical problems
Mistake 3: Using Inverse Functions on Invalid Values
Error: Trying to calculate sin⁻¹(1.5)
Reality: Sine values range from -1 to 1. A sine of 1.5 is impossible.
Cause: Usually an error in calculating the ratio. Double-check that you:
- Identified sides correctly
- Calculated the ratio correctly
- Used hypotenuse as the larger value in sine/cosine ratios
Mistake 4: Forgetting Which Sides Create Which Ratio
Error: Having opposite and adjacent, but using sin⁻¹ instead of tan⁻¹
Solution: Review SOH-CAH-TOA
- Hypotenuse involved? → Use sin⁻¹ or cos⁻¹
- Only the two legs? → Use tan⁻¹
Complete Triangle Solutions
Let's solve complete triangles using inverse functions.
Example: Complete Solution from Two Sides
Given: Right triangle with legs a = 5 and b = 12
Find: Hypotenuse c, angle A, angle B
Solution:
Step 1: Find hypotenuse (Pythagorean theorem)
- c² = 5² + 12² = 25 + 144 = 169
- c = 13
Step 2: Find angle A (opposite to side a)
Method using sine:
- sin(A) = 5/13 ≈ 0.385
- A = sin⁻¹(0.385) ≈ 22.62°
Method using tangent:
- tan(A) = 5/12 ≈ 0.417
- A = tan⁻¹(0.417) ≈ 22.62°
Both methods work!
Step 3: Find angle B
- B = 90° - 22.62° = 67.38°
Verification:
- All sides positive? ✓
- c is longest? ✓
- Pythagorean theorem satisfied? ✓
- Angles sum to 180°? 22.62° + 67.38° + 90° = 180° ✓
Special Values You Should Know
Memorizing these helps build intuition:
Inverse Sine
- sin⁻¹(0) = 0°
- sin⁻¹(0.5) = 30°
- sin⁻¹(0.707) ≈ 45° (actually 1/√2)
- sin⁻¹(0.866) ≈ 60° (actually √3/2)
- sin⁻¹(1) = 90°
Inverse Cosine
- cos⁻¹(1) = 0°
- cos⁻¹(0.866) ≈ 30°
- cos⁻¹(0.707) ≈ 45°
- cos⁻¹(0.5) = 60°
- cos⁻¹(0) = 90°
Inverse Tangent
- tan⁻¹(0) = 0°
- tan⁻¹(0.577) ≈ 30° (actually 1/√3)
- tan⁻¹(1) = 45°
- tan⁻¹(1.732) ≈ 60° (actually √3)
Pattern: Notice how sin⁻¹ and cos⁻¹ values are related due to complementary angles:
- sin⁻¹(0.5) = 30° and cos⁻¹(0.5) = 60° (complementary!)
Real-World Applications
Surveying and Navigation
Problem: You're 150 feet from a building. The top makes an angle of elevation such that the height is 80 feet. What's the angle?
Solution:
- tan(angle) = 80/150 ≈ 0.533
- angle = tan⁻¹(0.533) ≈ 28.07°
Construction
Problem: A roof rises 8 feet over a 20-foot span. What's the roof angle?
Solution:
- tan(angle) = 8/20 = 0.4
- angle = tan⁻¹(0.4) ≈ 21.8°
Engineering
Problem: A ramp has a 1:12 slope (rises 1 foot per 12 feet horizontal). What angle is this?
Solution:
- tan(angle) = 1/12 ≈ 0.0833
- angle = tan⁻¹(0.0833) ≈ 4.76°
This is the ADA-compliant maximum ramp angle.
The Inverse Function Relationship
The defining property of inverse functions:
sin(sin⁻¹(x)) = x (for -1 ≤ x ≤ 1)
sin⁻¹(sin(θ)) = θ (for -90° ≤ θ ≤ 90°)
What this means: If you apply a function and then its inverse (or vice versa), you get back what you started with—as long as you stay within the valid ranges.
Example:
- Start with: 0.5
- Apply sin⁻¹: sin⁻¹(0.5) = 30°
- Apply sin: sin(30°) = 0.5
- Back to start! ✓
Reverse order:
- Start with: 30°
- Apply sin: sin(30°) = 0.5
- Apply sin⁻¹: sin⁻¹(0.5) = 30°
- Back to start! ✓
This relationship is what makes them "inverse" functions.
Advanced Consideration: Multiple Solutions
In more advanced mathematics (beyond right triangles), you'll encounter situations where multiple angles have the same trig value.
Example: sin(30°) = sin(150°) = 0.5
For right triangles, this isn't an issue because angles are between 0° and 90°. But in other contexts, you need to consider which solution applies.
The inverse function always returns the principal value:
- sin⁻¹(0.5) returns 30° (not 150°)
- But 150° is also a valid angle with sine = 0.5
If you need the other angle, you calculate it using:
- Other angle = 180° - principal value
- For sin⁻¹: 180° - 30° = 150°
For right triangle work, just use the principal value—it's always correct.
Using Technology for Verification
When learning inverse trig functions, verifying your work helps build confidence. You can:
- Check that your angle produces the original ratio
- Verify angles sum correctly
- Confirm your triangle satisfies the Pythagorean theorem
Tools like a right triangle calculator can help verify your inverse trig calculations and show you if you've selected the correct function.
Practice Problems
Problem 1: In a right triangle, sin(A) = 0.6. Find angle A.
Solution
A = sin⁻¹(0.6) ≈ 36.87°
Problem 2: In a right triangle, opposite = 9, adjacent = 12. Find the angle.
Solution
tan(θ) = 9/12 = 0.75; θ = tan⁻¹(0.75) ≈ 36.87°
Problem 3: In a right triangle, adjacent = 7, hypotenuse = 10. Find the angle.
Solution
cos(θ) = 7/10 = 0.7; θ = cos⁻¹(0.7) ≈ 45.57°
Problem 4: If angle A = 40°, what is angle B in a right triangle?
Solution
B = 90° - 40° = 50° (No inverse trig needed!)
The Complete Picture: Forward and Inverse
Together, regular and inverse trigonometric functions give you complete control over right triangles:
Have angles, need sides?
→ Use sin, cos, tan (forward direction)
Have sides, need angles?
→ Use sin⁻¹, cos⁻¹, tan⁻¹ (inverse direction)
Have some of each?
→ Use whichever tools fit what you know
This completeness is what makes trigonometry so powerful. No right triangle problem is beyond your reach once you master both directions.
Conclusion: Completing the Trigonometric Toolkit
Inverse trigonometric functions aren't just mathematical curiosities—they're essential tools that complete your ability to solve triangle problems. Regular trig functions (sin, cos, tan) work forward from angles to ratios. Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) work backward from ratios to angles.
Together, they form a complete system:
Forward: "I know the angle is 30°. What's the sine?" → sin(30°) = 0.5
Inverse: "I know the sine is 0.5. What's the angle?" → sin⁻¹(0.5) = 30°
This bidirectional capability transforms trigonometry from a one-way street into a complete navigation system for triangles. You can start anywhere—with angles, with sides, with ratios—and find your way to complete solutions.
The notation might seem intimidating at first. The restricted ranges might seem arbitrary. But these tools follow logical mathematical principles designed to give you unique, usable answers.
Master inverse trigonometric functions, and you'll never be stuck with a triangle problem again. Know the sides? Find the angles. Know one angle? Find the others. Know mixed information? Piece together the complete picture.
That's the power of inverse functions—they complete the story, working backward to find where you started, turning the known into discovery paths for the unknown.
From ratio to angle, from measurement to geometry, inverse trigonometric functions are your bridge back across the mathematical river. Learn to use them well, and every triangle becomes an open book, its angles and sides yours to calculate at will.
