Trigonometry Calculator — Sin, Cos, Tan & Inverse Functions
Calculate all six trigonometric functions (sin, cos, tan, csc, sec, cot) for any angle in degrees or radians. Includes inverse trig functions and a unit circle reference. See also Right Triangle Calculator and Pythagorean Theorem Calculator.
Trigonometric Functions Calculator
Inverse Trigonometric Functions
Enter a value to find the angle. arcsin and arccos accept values between −1 and 1.
How to Use the Trigonometry Calculator
Enter any angle in degrees or radians and click Calculate to see all six trigonometric function values at once. Use the quick angle buttons for common angles like 30°, 45°, 60°, and 90°. The calculator handles all four quadrants and special angles. For inverse calculations, select arcsin, arccos, or arctan, enter a ratio value, and get the corresponding angle in both degrees and radians.
Trigonometric Function Formulas
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent = sin θ / cos θ
csc θ = 1 / sin θ = hypotenuse / opposite
sec θ = 1 / cos θ = hypotenuse / adjacent
cot θ = 1 / tan θ = cos θ / sin θ = adjacent / opposite
Pythagorean identity: sin²θ + cos²θ = 1
Radians to degrees: degrees = radians × (180 / π)
Degrees to radians: radians = degrees × (π / 180)
Example Calculation
Find all trig functions for θ = 45°:
45° = π/4 radians ≈ 0.7854 rad
sin 45° = √2/2 ≈ 0.7071
cos 45° = √2/2 ≈ 0.7071
tan 45° = 1
csc 45° = √2 ≈ 1.4142
sec 45° = √2 ≈ 1.4142
cot 45° = 1
Trigonometric Values Reference Table
| Angle (°) | Radians | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | undef | 1 | undef |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | π/2 | 1 | 0 | undef | 1 | undef | 0 |
Solved Examples
Example 1: Ladder Against a Wall
A 10-meter ladder leans against a wall making a 65° angle with the ground. How high up the wall does the ladder reach?
The height is the side opposite the 65° angle, and the ladder is the hypotenuse.
sin 65° = opposite / hypotenuse = height / 10
height = 10 × sin 65°
height = 10 × 0.9063
height = 9.063 meters
Example 2: Building Height from Distance and Elevation Angle
You stand 50 meters from the base of a building. The angle of elevation to the top is 32°. How tall is the building?
The height is opposite the angle, and the distance is adjacent.
tan 32° = opposite / adjacent = height / 50
height = 50 × tan 32°
height = 50 × 0.6249
height = 31.24 meters
Example 3: Finding an Unknown Trig Value Using the Pythagorean Identity
If sin θ = 3/5 and θ is in the first quadrant, find cos θ.
Use the identity: sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 1 - 9/25 = 16/25
cos θ = 4/5 (positive in first quadrant)
cos θ = 0.8
Example 4: Using Inverse Trig to Find an Angle
A ramp rises 3 meters over a horizontal distance of 8 meters. What angle does the ramp make with the ground?
tan θ = opposite / adjacent = 3 / 8 = 0.375
θ = arctan(0.375)
θ = 20.56°
Practice Questions
1. A kite string is 80 meters long and makes a 40° angle with the ground. How high is the kite above the ground?
Answer: height = 80 × sin 40° = 80 × 0.6428 = 51.42 meters
2. Find tan θ if sin θ = 5/13 and θ is in the first quadrant.
Answer: cos θ = 12/13 (from Pythagorean identity), so tan θ = sin θ / cos θ = (5/13) / (12/13) = 5/12 = 0.4167
3. Convert 225° to radians and find sin 225°.
Answer: 225° × (π/180) = 5π/4 radians. sin 225° = -√2/2 = -0.7071 (third quadrant, sin is negative)
4. A surveyor measures the angle of depression from the top of a 120-meter cliff to a boat as 28°. How far is the boat from the base of the cliff?
Answer: tan 28° = 120 / distance, so distance = 120 / tan 28° = 120 / 0.5317 = 225.7 meters
5. Find the angle whose cosine is 0.6.
Answer: θ = arccos(0.6) = 53.13°
6. If sec θ = 2, find sin θ (first quadrant).
Answer: sec θ = 1/cos θ = 2, so cos θ = 1/2. Then sin²θ = 1 - (1/2)² = 3/4, so sin θ = √3/2 = 0.8660
Common Mistakes to Avoid
Using the wrong angle mode (degrees vs. radians)
The most frequent error in trigonometry. If your calculator is set to radians and you enter 45 expecting degrees, you get sin(45 rad) = 0.8509 instead of sin(45°) = 0.7071. Always verify your mode before computing.
Mixing up SOH-CAH-TOA ratios
A common mistake is confusing which sides go with which function. Remember: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Drawing and labeling the triangle relative to the angle of interest prevents this error.
Forgetting that some trig values are undefined
tan 90° and tan 270° are undefined (division by zero). Similarly, csc 0°, csc 180°, sec 90°, sec 270°, cot 0°, and cot 180° are all undefined. Attempting to evaluate these leads to errors or infinity.
Key Takeaways
- The six trig functions (sin, cos, tan, csc, sec, cot) relate angles to side ratios in right triangles and extend to all angles via the unit circle.
- Always check whether your angle is in degrees or radians before calculating -- this is the single most common source of error.
- The Pythagorean identity sin²θ + cos²θ = 1 lets you find one function value from another without needing the triangle.
- Inverse trig functions (arcsin, arccos, arctan) reverse the process: given a ratio, they return the angle.
- Trig functions repeat every 360° (or 2π radians), and their signs depend on the quadrant of the angle.
- In real-world problems, always identify which side is opposite, adjacent, or the hypotenuse relative to the angle you are working with.
Frequently Asked Questions
What is the difference between degrees and radians?
Degrees and radians are two units for measuring angles. A full circle is 360° or 2π radians. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Radians are the standard unit in calculus and most mathematical formulas.
Why is tan 90° undefined?
tan θ = sin θ / cos θ. At 90°, cos 90° = 0, so you would be dividing by zero. Geometrically, the tangent line at 90° on the unit circle is vertical and extends to infinity. The same applies to cot 0°, csc 0°, and sec 90°.
What are the reciprocal trig functions?
The three reciprocal functions are cosecant (csc = 1/sin), secant (sec = 1/cos), and cotangent (cot = 1/tan). They are useful in calculus, physics, and engineering for simplifying expressions where division by sin, cos, or tan appears frequently.
What are inverse trig functions used for?
Inverse trig functions (arcsin, arccos, arctan) find the angle when you know the ratio. For example, if sin θ = 0.5, then arcsin(0.5) = 30°. They are essential for solving triangles, navigation, physics problems, and converting between rectangular and polar coordinates.
What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin. For any angle θ, the point on the unit circle is (cos θ, sin θ). It provides a geometric way to understand all trig functions and is the foundation for extending trig beyond right triangles to all real numbers.