Trigonometry Solver
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Result
Calculates all six trig functions: sin, cos, tan, csc, sec, cot.
arcsin and arccos: value must be between −1 and 1.
C = 90° always. Fill in only the known values shown below.
Common angles: 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330, 360
Shows all 6 cofunction pairs and their numeric values.
Quick Examples
Complete Guide to Trigonometry: Every Topic Covered Step by Step
From understanding sin, cos, and tan to solving triangles, applying identities, and mastering the unit circle — everything students need from middle school through college.
How to Use This Trigonometry Solver
Click Trig Fns. Enter any angle and choose degrees or radians. Press Calculate to see all six functions at once.
For Inverse: choose arcsin, arccos, or arctan, enter the ratio value, and get the angle in both degrees and radians.
Right Triangle: pick what you know from the dropdown and fill in just those values. Law of Sines and Law of Cosines: select the case (AAS, SSA, SAS, or SSS) and enter your numbers.
Every result shows all sides, all angles, and the area.
Unit Circle: type any angle to get coordinates, sin, cos, tan, and the reference angle. Identities: choose the identity type, enter the angle or value, and see the formula applied step by step.
Cofunction: enter any angle to see all six cofunction pairs.
What Is Trigonometry?
What is trigonometry and why does it matter?
Trigonometry is the branch of mathematics that studies the relationships between angles and side lengths in triangles. It started as a tool for astronomy and navigation and now appears in physics, engineering, architecture, computer graphics, and music. Any time you need to find a missing distance using an angle, trigonometry is the tool you reach for.
What is SOH-CAH-TOA?
SOH-CAH-TOA is the most important memory aid in trigonometry. SOH: sin = Opposite / Hypotenuse. CAH: cos = Adjacent / Hypotenuse. TOA: tan = Opposite / Adjacent. These three ratios describe how the sides of a right triangle relate to one of its acute angles. Every right triangle problem starts here.
What is the difference between degrees and radians?
Both measure angles but in different units. Degrees divide a full circle into 360 parts. Radians divide it into 2π parts (about 6.283). The conversion is: radians = degrees × π/180. For example, 90° = π/2 radians. This calculator accepts both. Switch the unit selector in the Trig Fns tab to choose.
The Six Trigonometric Functions (Trig Fns Tab)
What are the six trigonometric functions?
The six functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Sin, cos, and tan are the primary three. Csc = 1/sin, sec = 1/cos, cot = 1/tan. The three reciprocal functions are just the flipped versions of the primary three.
What are the exact values of sin and cos at standard angles?
At 0°: sin = 0, cos = 1. At 30°: sin = 1/2, cos = √3/2. At 45°: sin = √2/2, cos = √2/2. At 60°: sin = √3/2, cos = 1/2. At 90°: sin = 1, cos = 0. Memorizing these five angles covers most exam questions. The Trig Fns tab computes exact decimal values for any angle you enter.
When are trig functions undefined?
Tan and sec are undefined when cos = 0, which happens at 90°, 270°, and their equivalents. Csc and cot are undefined when sin = 0, which happens at 0°, 180°, 360°. The calculator displays "undefined" for these cases automatically rather than showing an error.
How do trig values change across the four quadrants?
Quadrant I (0° to 90°): all positive. Quadrant II (90° to 180°): sin positive, cos and tan negative. Quadrant III (180° to 270°): tan positive, sin and cos negative. Quadrant IV (270° to 360°): cos positive, sin and tan negative. The memory aid "All Students Take Calculus" gives you the order: All, Sin, Tan, Cos.
Inverse Trigonometric Functions (Inverse Tab)
What is arcsin?
arcsin(x) is the angle whose sine equals x. If sin(30°) = 0.5, then arcsin(0.5) = 30°. The input must be between -1 and 1 because sine never goes outside that range. The output is always between -90° and 90°. Use the Inverse tab with arcsin selected to calculate this directly.
What is arccos?
arccos(x) is the angle whose cosine equals x. If cos(60°) = 0.5, then arccos(0.5) = 60°. The input must be between -1 and 1. The output is always between 0° and 180°. Arccos is used when you know the adjacent and hypotenuse and want the angle.
What is arctan?
arctan(x) is the angle whose tangent equals x. Unlike arcsin and arccos, arctan accepts any real number as input because tan can be any value. The output is always between -90° and 90°. If you know two legs of a right triangle, arctan gives you the angle: A = arctan(opposite/adjacent).
Solve: find the angle if sin(A) = 0.866
Enter arcsin in the Inverse tab and type 0.866. The result is A ≈ 60°, which is 1.047 radians. Verify: sin(60°) = √3/2 ≈ 0.866. This is correct. Inverse trig functions are the key tool for finding angles when you know a side ratio.
Solving Right Triangles (Right △ Tab)
What information do you need to solve a right triangle?
You need any two pieces of information other than the right angle itself. That could be two sides, or one side and one acute angle. With any two knowns, you can find all three sides and both acute angles. The Right Triangle tab lets you pick what you know from a dropdown and fills in the rest automatically.
Solve: a right triangle with legs a = 3 and b = 4
Step 1: hypotenuse c = √(3² + 4²) = √25 = 5. Step 2: angle A = arctan(3/4) = arctan(0.75) ≈ 36.87°. Step 3: angle B = 90° - 36.87° = 53.13°. Area = ½ × 3 × 4 = 6. Select "Two legs a and b" in the Right Triangle tab, enter 3 and 4, and verify.
How do you solve a right triangle given hypotenuse and one angle?
If c = 10 and A = 30°: then a = c × sin(A) = 10 × 0.5 = 5. And b = c × cos(A) = 10 × 0.866 = 8.66. Angle B = 90° - 30° = 60°. Select "Hypotenuse c and angle A" in the dropdown and enter those two values to confirm.
What is the Pythagorean theorem and how does it connect to trigonometry?
The Pythagorean theorem states a² + b² = c² for any right triangle. It is the special case of the Law of Cosines when C = 90° (since cos 90° = 0). It also gives the Pythagorean identity sin²θ + cos²θ = 1, because on the unit circle a = sin, b = cos, and c = 1.
Law of Sines (Law of Sines Tab)
What is the Law of Sines?
The Law of Sines states that in any triangle: a / sin(A) = b / sin(B) = c / sin(C). This ratio is constant for a given triangle and equals the diameter of its circumscribed circle. You use it whenever you know two angles and any side (AAS or ASA), or two sides and an angle not between them (SSA).
How do you use the Law of Sines to solve a triangle?
Set up the ratio with your known pair. For A = 45°, a = 10, B = 60°: first find C = 180° - 45° - 60° = 75°. Then b = a × sin(B) / sin(A) = 10 × sin(60°) / sin(45°) = 10 × 0.866 / 0.707 = 12.25. Then c = a × sin(C) / sin(A) = 10 × sin(75°) / sin(45°) = 13.66. Enter these values in the Law of Sines tab to verify.
What is the ambiguous case (SSA)?
When you know two sides and an angle not between them (SSA), there can be zero, one, or two valid triangles. If sin(B) = b × sin(A) / a is greater than 1, no triangle exists. If it equals 1 exactly, there is one right triangle. If it is less than 1, there may be two triangles. The calculator handles the primary solution and flags invalid cases.
Law of Cosines (Law of Cosines Tab)
What is the Law of Cosines?
The Law of Cosines states c² = a² + b² − 2ab × cos(C). It is used when you know two sides and the included angle (SAS) or all three sides (SSS). It is a generalization of the Pythagorean theorem. When C = 90°, cos(90°) = 0 and the formula becomes the familiar a² + b² = c².
How do you solve an SAS triangle?
Given a = 5, b = 7, C = 60°: find c = √(25 + 49 - 2 × 5 × 7 × cos(60°)) = √(74 - 35) = √39 ≈ 6.24. Then use the Law of Sines to find A and B. Select SAS in the Law of Cosines tab and enter those three values to confirm all angles and the area.
How do you find all angles from three sides (SSS)?
Rearrange the formula: cos(C) = (a² + b² − c²) / (2ab). For a = 3, b = 4, c = 5: cos(C) = (9 + 16 − 25) / 24 = 0 / 24 = 0, so C = arccos(0) = 90°. This confirms the 3-4-5 right triangle. Select SSS in the Law of Cosines tab and enter 3, 4, 5 to see all three angles calculated step by step.
The Unit Circle (Unit Circle Tab)
What is the unit circle?
The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. Every point on it has coordinates (cosθ, sinθ), where θ is the angle measured counterclockwise from the positive x-axis. It is the foundation for extending trig functions beyond right triangles to any angle.
What are the coordinates at standard angles on the unit circle?
At 0°: (1, 0). At 30°: (√3/2, 1/2). At 45°: (√2/2, √2/2). At 60°: (1/2, √3/2). At 90°: (0, 1). At 180°: (-1, 0). At 270°: (0, -1). Knowing these by memory makes any trig exam significantly faster. The Unit Circle tab shows the coordinates and trig values for any angle you enter.
What is a reference angle?
A reference angle is the acute angle (between 0° and 90°) between the terminal side of your angle and the x-axis. It lets you use the first-quadrant values for any angle in any quadrant. For 150°: the reference angle is 180° - 150° = 30°. For 225°: it is 225° - 180° = 45°. Then apply the correct sign based on the quadrant.
Trigonometric Identities (Identities Tab)
What are the Pythagorean identities?
There are three: sin²θ + cos²θ = 1 (the most important), 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ. All three come directly from the Pythagorean theorem applied to the unit circle. The Identities tab uses the first one to find all trig values when you know just one of them.
What are the double angle formulas?
sin(2θ) = 2 sin(θ) cos(θ). cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1. tan(2θ) = 2tan(θ) / (1 − tan²θ). For θ = 30°: sin(60°) = 2 × 0.5 × 0.866 = 0.866. Select Double Angle in the Identities tab and enter 30 to see the full working.
What are the half angle formulas?
sin(θ/2) = ±√((1 − cosθ) / 2). cos(θ/2) = ±√((1 + cosθ) / 2). tan(θ/2) = sinθ / (1 + cosθ). The sign depends on the quadrant that θ/2 lands in. Select Half Angle in the Identities tab to compute these for any angle.
What are sum and difference identities?
sin(A + B) = sin A × cos B + cos A × sin B. cos(A + B) = cos A × cos B − sin A × sin B. For subtraction, flip the sign of the second term in each. These let you compute exact values like sin(75°) = sin(45° + 30°) without a calculator. The Sum and Difference options in the Identities tab show every multiplication step.
Cofunction Identities (Cofunction Tab)
What are cofunction identities?
Cofunction identities state that each trig function of an angle equals its cofunction of the complementary angle (90° minus that angle): sin(θ) = cos(90° − θ), cos(θ) = sin(90° − θ), tan(θ) = cot(90° − θ), csc(θ) = sec(90° − θ), sec(θ) = csc(90° − θ), cot(θ) = tan(90° − θ).
Why do cofunction identities work?
In a right triangle, the two acute angles always add up to 90°. If one angle is θ, the other is 90° − θ. The opposite side for θ is the adjacent side for 90° − θ. This is exactly why sin(θ) = cos(90° − θ) — they share the same side, just described differently from each angle.
How do you use cofunction identities on exams?
They let you rewrite expressions to simplify proofs and equations. If you see cos(60°) in a problem, you can rewrite it as sin(30°) and use a memorized value. They also appear in limits: lim as x approaches 0 of sin(x)/x = 1 uses the fact that sin and cos are cofunctions. The Cofunction tab shows all six pairs for any angle you enter.
Trigonometry by Student Level
Start with: Trig Fns tab at 30°, 45°, 60°. Right Triangle tab with two legs known. Memorize SOH-CAH-TOA before anything else.
How to use: Enter a standard angle, see all six values, then sketch the triangle by hand to connect the numbers to the geometry.
Tip: Use the Quick Examples buttons to explore different angles without typing. Observe how the values change as the angle increases.
Focus on: Law of Sines and Law of Cosines for non-right triangles. Unit Circle for reference angles. Identities for proofs and simplification.
How to use: Work the problem by hand, then enter it in the calculator to verify. Pay close attention to the sign rules in Quadrants II, III, and IV.
Tip: Use the Identities tab with Double Angle and Sum/Difference to verify identity proofs before writing them up in homework.
Key uses: Cofunction and Pythagorean identities in calculus proofs. Law of Cosines in vectors and physics. Inverse trig in integration (trig substitution).
How to use: Use the calculator as a quick numerical check during derivations. The Unit Circle tab confirms reference angle signs mid-calculation.
Tip: Enter the Half Angle identity for angles that appear in integration by parts to double-check the substitution values.
Using This Tool for Trig Assignments
Homework Problems
Work each problem by hand first. Then enter it in the matching tab to verify. If your answer differs, compare the step-by-step output line by line to find where the error happened. The most common homework mistakes are using the wrong quadrant sign and mixing up which side is opposite versus adjacent.
Quizzes and Tests
Press Random 10 to 15 times before a quiz. Solve each example on paper in under 90 seconds, then check. Timed practice across all eight tabs builds the pattern recognition that trig tests reward. Focus your extra rounds on whichever tab produces the most errors.
Word Problems and Case Studies
Identify what type of triangle the problem describes. If it mentions a right angle, use the Right Triangle tab. If it gives you two sides and a non-included angle, use Law of Sines. If it gives three sides, use Law of Cosines. Map the problem to the correct tab before entering any numbers.
Problem Sets and Proofs
For multi-step problem sets, verify each trig value before substituting it into the next step. A wrong sin(A) in step one will break every calculation that follows. For identity proofs, use the Identities tab to check that the left side and right side evaluate to the same decimal for several test angles.
Frequently Asked Questions
- 1. What are the six trigonometric functions?
- The six functions are sin, cos, tan, csc, sec, and cot. Sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Csc, sec, and cot are the reciprocals of sin, cos, and tan. The Trig Fns tab computes all six for any angle you enter.
- 2. How do you find an angle if you know a trig ratio?
- Use an inverse trig function. If sin(A) = 0.5, then A = arcsin(0.5) = 30°. If cos(A) = 0.707, then A = arccos(0.707) = 45°. If tan(A) = 1, then A = arctan(1) = 45°. The Inverse tab handles all three inverse functions and returns the angle in both degrees and radians.
- 3. How do you solve a right triangle?
- You need any two values (sides or angles). Use the Pythagorean theorem for sides: c = √(a² + b²). Use SOH-CAH-TOA for angles: A = arctan(a/b) or arcsin(a/c) or arccos(b/c). The Right Triangle tab accepts any two knowns from a dropdown and solves all five remaining values automatically.
- 4. What is the Law of Sines and when do you use it?
- The Law of Sines is a/sin(A) = b/sin(B) = c/sin(C). Use it for non-right triangles when you know: two angles and any side (AAS or ASA), or two sides and a non-included angle (SSA). The SSA case can produce zero, one, or two triangles (the ambiguous case). The Law of Sines tab handles all these cases.
- 5. What is the Law of Cosines and when do you use it?
- The Law of Cosines is c² = a² + b² − 2ab·cos(C). Use it when you know two sides and the included angle (SAS) or all three sides (SSS). It is more powerful than the Law of Sines for these cases because it avoids the ambiguous case. The Law of Cosines tab handles both SAS and SSS directly.
- 6. What is the unit circle and how do you use it?
- The unit circle is a circle of radius 1 where every point is (cosθ, sinθ). It extends trig functions to all angles, not just those in right triangles. Memorize the coordinates at 0°, 30°, 45°, 60°, 90°, and their equivalents in other quadrants. The Unit Circle tab shows the exact coordinates, trig values, quadrant, and reference angle for any input.
- 7. What is sin²θ + cos²θ = 1 used for?
- This Pythagorean identity is used to find one trig value when you know another. If sin(θ) = 0.6, then cos²(θ) = 1 − 0.36 = 0.64, so cos(θ) = 0.8 (in Quadrant I). It also appears in calculus for substitutions and in proving other identities. The Pythagorean option in the Identities tab shows all six values from any one input.
- 8. How do double angle formulas help on exams?
- Double angle formulas let you compute trig values of double angles without a table. sin(60°) = sin(2×30°) = 2·sin(30°)·cos(30°) = 2×0.5×0.866 = 0.866. They also appear in integration (power-reducing formulas) and in proving other identities. Select Double Angle in the Identities tab to see the step-by-step application for any angle.
- 9. What are cofunction identities?
- Cofunction identities relate each trig function to its complement: sin(θ) = cos(90°−θ), tan(θ) = cot(90°−θ), sec(θ) = csc(90°−θ). They work because in a right triangle, the two acute angles sum to 90° and share the same set of sides. The Cofunction tab shows all six pairs and their values for any angle.
- 10. Is this trigonometry solver useful for AP, IB, and GCSE exams?
- Yes. The Law of Sines and Law of Cosines tabs cover all non-right triangle problems that appear on AP Precalculus and AP Calculus BC. The Identities tab covers proof-based questions on IB Math HL and GCSE Further Math. Use the calculator to verify hand-written work before submitting. Always work the problem yourself first, then check with the solver.