Trigonometry Identities & Unit Circle
All the trigonometry identities, special angle values and the unit circle on one printable page for quick revision.
Unit circle
Each point is (cos θ, sin θ). The x-coordinate is cosine and the y-coordinate is sine.
Special angle values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| radians | 0 | π/6 | π/4 | π/3 | π/2 |
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
| csc θ | undefined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | undefined |
| cot θ | undefined | √3 | 1 | 1/√3 | 0 |
Degrees to radians: rad = deg × π/180. Radians to degrees: deg = rad × 180/π. Signs by quadrant (ASTC): I all positive, II sine, III tangent, IV cosine.
Basic identities
| Type | Identities |
|---|---|
| Reciprocal | csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ |
| Quotient | tan θ = sin θ/cos θ, cot θ = cos θ/sin θ |
| Pythagorean | sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ |
| Negative angle | sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ |
| Complementary | sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ |
Sum and difference
| sin(A ± B) | sin A cos B ± cos A sin B |
| cos(A ± B) | cos A cos B ∓ sin A sin B |
| tan(A ± B) | (tan A ± tan B) / (1 ∓ tan A tan B) |
Double, triple and half angle
| sin 2A | 2 sin A cos A |
| cos 2A | cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A |
| tan 2A | 2 tan A / (1 − tan²A) |
| sin 3A | 3 sin A − 4 sin³A |
| cos 3A | 4 cos³A − 3 cos A |
| sin²A, cos²A | (1 − cos 2A)/2, (1 + cos 2A)/2 |
| sin(A/2), cos(A/2) | ±√((1 − cos A)/2), ±√((1 + cos A)/2) |
| tan(A/2) | sin A/(1 + cos A) = (1 − cos A)/sin A |
Product-to-sum and sum-to-product
| 2 sin A cos B | sin(A + B) + sin(A − B) |
| 2 cos A sin B | sin(A + B) − sin(A − B) |
| 2 cos A cos B | cos(A + B) + cos(A − B) |
| 2 sin A sin B | cos(A − B) − cos(A + B) |
| sin C + sin D | 2 sin((C + D)/2) cos((C − D)/2) |
| sin C − sin D | 2 cos((C + D)/2) sin((C − D)/2) |
| cos C + cos D | 2 cos((C + D)/2) cos((C − D)/2) |
| cos C − cos D | −2 sin((C + D)/2) sin((C − D)/2) |
Triangles
| Law of sines | a/sin A = b/sin B = c/sin C = 2R |
| Law of cosines | c² = a² + b² − 2ab cos C |
| Area | ½ ab sin C |
FAQ
How do I remember the unit circle?
Remember the first quadrant values for 30, 45 and 60 degrees, then use symmetry: the other quadrants have the same values with different signs (ASTC: All, Sine, Tangent, Cosine).
What is sin squared plus cos squared?
It always equals 1. This is the Pythagorean identity and the other two are found by dividing it by cos squared or sin squared.
How do I convert degrees to radians?
Multiply by pi and divide by 180. For example 60 degrees is pi/3.
Can I print this page?
Yes. Press Print / Save as PDF. The page switches to a clean black-on-white layout without menus.
Where are tan 90 and cot 0 values?
They are undefined because the denominator (cos 90 or sin 0) is zero.