Trigonometry, with the degrees-or-radians question settled first
Evaluate trig functions in whichever angle mode the question is set in, solve a right triangle from any two values, and check the unit circle and identities without opening a different tab. The mode switch is at the top because getting it wrong is the single most common reason a trig answer looks inexplicably off.
For derivatives of trig functions, the calculus solver handles them; for drawing the curves, use the graphing calculator.
Degrees or radians: decide before you calculate
Almost every confusing trig answer traces back to angle mode. In degrees sin(30) = 0.5; in radians the same expression means 30 radians and gives about -0.988. Neither is wrong, because they answer different questions, but a calculator that hides which one it is using will quietly produce nonsense.
The rule of thumb: geometry, surveying and most school exercises are in degrees. Anything involving calculus, π, or the unit circle is in radians, because the derivative rules only take their familiar form there. d/dx(sin x) = cos x holds in radians and nowhere else.
Degrees
sin(30) = 0.5
A full turn is 360°. Use for triangles, bearings, and most problems written with a ° symbol.
Radians
cos(π) = −1
A full turn is 2π. Use this for calculus and anything written in terms of π, because the derivative rules assume it.
Inverse functions
sin⁻¹(0.5) = 30°
Returns the angle in whichever mode is selected. Only defined for inputs between −1 and 1 for sin and cos.
Where tan breaks
tan(90°) undefined
tan = sin/cos, and cos is zero at 90° and 270°, so the division is impossible. Reported as undefined rather than a huge number.
Two triangles, worked through
Two sides known
opposite 3, adjacent 4- Both legs are known, so Pythagoras gives the hypotenuse: h = √(3² + 4²) = √25 = 5.
- For the angle, the two known sides are opposite and adjacent, which is tangent (TOA): tan A = 3/4.
- So A = tan⁻¹(0.75) ≈ 36.87°.
- The third angle follows from the angle sum: 90° − 36.87° = 53.13°.
A side and an angle
hypotenuse 10, angle 30°- Opposite comes from sine (SOH): o = 10 × sin(30°) = 10 × 0.5 = 5.
- Adjacent comes from cosine (CAH): a = 10 × cos(30°) ≈ 8.66.
- Check with Pythagoras: 5² + 8.66² ≈ 100 = 10² ✓
- This is the 30-60-90 triangle, where the side opposite 30° is always exactly half the hypotenuse.
Give it any two, get the rest
Enter any two of the three sides and the angle, and the remaining values are worked out with the rule that applies: Pythagoras for a missing side, SOH-CAH-TOA for a missing angle. Angles here are always in degrees, because a triangle question is a geometry question.
The values worth knowing by heart
These are the angles that turn up constantly, and the exact values are expected rather than decimals. The pattern is easier to remember than the table: the numerators of sine run 0, 1, √2, √3, 2 over 2, and cosine runs the same sequence backwards.
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
tan is undefined at 90° and 270° because cos is zero there, and tan = sin/cos.
The ones you actually need
Pythagorean
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
The first one is just Pythagoras applied to a radius-1 triangle. Almost every simplification starts here.
Quotient & reciprocal
tanθ = sinθ / cosθ
csc = 1/sin, sec = 1/cos, cot = 1/tan
Note that sec pairs with cos rather than sin. The naming is counterintuitive and catches nearly everyone once.
Double angle
sin2θ = 2 sinθ cosθ
cos2θ = cos²θ − sin²θ
The cosine version has three equivalent forms; this one converts to the others using the Pythagorean identity.
Angle sum
sin(A+B) = sinA cosB + cosA sinB
cos(A+B) = cosA cosB − sinA sinB
Watch the sign on cosine. It flips relative to what the pattern suggests.
Trigonometry FAQ
Why does cos(pi) give 0.998 on some calculators?
Because they are in degrees and are reading π as the number 3.14159 degrees, not as half a turn. cos(3.14159°) really is about 0.9985. Switch to radians and you get −1. This is the most common trig calculator mistake, which is why the mode switch here is at the top rather than buried in a settings menu.
When should I use radians instead of degrees?
Any time calculus is involved, or the problem is written in terms of π. The derivative rules, d/dx(sin x) = cos x among them, are only true in radians. Degrees are fine for triangle geometry and most school exercises.
What is SOH-CAH-TOA?
A mnemonic for which ratio to use: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Pick whichever one involves the two sides you know and the one you want.
Why is tan undefined at 90 degrees?
Because tan = sin/cos and cos(90°) = 0, so it asks you to divide by zero. As the angle approaches 90° the value grows without limit, but at exactly 90° there is no value, so the tool says undefined rather than returning a very large number.
How many values do I need to solve a right triangle?
Two, as long as at least one is a side. Two sides give you everything through Pythagoras and an inverse trig function; one side plus the angle gives the rest through sine and cosine. Two angles alone only fix the shape, not the size.
Does it solve trig equations like sin(x) = 0.5?
Not yet. The evaluator works on expressions with known angles, and the triangle solver works on triangles. For sin(x) = 0.5 you would take the inverse: sin⁻¹(0.5) = 30°, remembering that 150° also works, and that every solution repeats every 360°.