The six ratios (right-angled triangle)
For an angle θ in a right-angled triangle: opposite is the side across from θ, adjacent is the side next to it.
sin θopposite / hypotenuse
cos θadjacent / hypotenuse
tan θopposite / adjacenttan θ = sin θ / cos θ
cosec θhypotenuse / oppositecosec = 1 / sin
sec θhypotenuse / adjacentsec = 1 / cos
cot θadjacent / oppositecot = 1 / tan = cos / sin
Standard angle values
Learn these five rows - most school and exam questions use them.
θ = 0°sin 0, cos 1, tan 0
θ = 30°sin 1/2, cos √3/2, tan 1/√3
θ = 45°sin √2/2, cos √2/2, tan 1
θ = 60°sin √3/2, cos 1/2, tan √3
θ = 90°sin 1, cos 0, tan undefined
Pythagorean identities
Basicsin²θ + cos²θ = 1
Divided by cos²1 + tan²θ = sec²θ
Divided by sin²1 + cot²θ = cosec²θ
Co-function (complementary angles)
sin(90° - θ)= cos θ
cos(90° - θ)= sin θ
tan(90° - θ)= cot θ
cosec(90° - θ)= sec θ
sec(90° - θ)= cosec θ
cot(90° - θ)= tan θ
Signs by quadrant
Memory aid: All Students Take Coffee - All, Sin, Tan, Cos.
Quadrant I (0-90°)all six ratios positive
Quadrant II (90-180°)sin and cosec positive
Quadrant III (180-270°)tan and cot positive
Quadrant IV (270-360°)cos and sec positive
Negative angles
sin(-θ)= -sin θ
cos(-θ)= cos θ
tan(-θ)= -tan 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 Bminus for +, plus for -
tan(A ± B)(tan A ± tan B) / (1 ∓ tan A tan B)
Double angle
sin 2θ2 sin θ cos θ
cos 2θcos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan 2θ2 tan θ / (1 - tan²θ)
Laws for any triangle
For triangles with no right angle: sides a, b, c sit opposite angles A, B, C.
Sine rulea / sin A = b / sin B = c / sin C
Cosine rulec² = a² + b² - 2ab·cos C
Degrees and radians
Conversion180° = π radians
Degrees to radiansmultiply by π/180
Radians to degreesmultiply by 180/π
Common values30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
