Trigonometry

Basic Trig Ratios

sin θ = opp / hyp—SOH

cos θ = adj / hyp—CAH

tan θ = opp / adj—TOA

csc θ = 1 / sin θ—Cosecant

sec θ = 1 / cos θ—Secant

cot θ = 1 / tan θ—Cotangent

Unit Circle — Q1 & Q2

0° (0): sin=0, cos=1—

30° (π/6): sin=1/2, cos=√3/2—

45° (π/4): sin=√2/2, cos=√2/2—

60° (π/3): sin=√3/2, cos=1/2—

90° (π/2): sin=1, cos=0—

120° (2π/3): sin=√3/2, cos=-1/2—

135° (3π/4): sin=√2/2, cos=-√2/2—

150° (5π/6): sin=1/2, cos=-√3/2—

180° (π): sin=0, cos=-1—

Unit Circle — Q3 & Q4

210° (7π/6): sin=-1/2, cos=-√3/2—

225° (5π/4): sin=-√2/2, cos=-√2/2—

240° (4π/3): sin=-√3/2, cos=-1/2—

270° (3π/2): sin=-1, cos=0—

300° (5π/3): sin=-√3/2, cos=1/2—

315° (7π/4): sin=-√2/2, cos=√2/2—

330° (11π/6): sin=-1/2, cos=√3/2—

360° (2π): sin=0, cos=1—

Pythagorean Identities

sin²θ + cos²θ = 1—Fundamental identity

1 + tan²θ = sec²θ—Divide by cos²θ

1 + cot²θ = csc²θ—Divide by sin²θ

sin²θ = (1 - cos 2θ)/2—Power-reducing

cos²θ = (1 + cos 2θ)/2—Power-reducing

Double & Half Angle

sin 2θ = 2 sin θ cos θ—Double angle sine

cos 2θ = cos²θ - sin²θ—Double angle cosine

cos 2θ = 2cos²θ - 1—Alternate form

cos 2θ = 1 - 2sin²θ—Alternate form

tan 2θ = 2tanθ/(1-tan²θ)—Double angle tangent

sin(θ/2) = ±√((1-cosθ)/2)—Half angle sine

cos(θ/2) = ±√((1+cosθ)/2)—Half angle cosine

Sum & Difference

sin(A±B) = sinA cosB ± cosA sinB—

cos(A±B) = cosA cosB ∓ sinA sinB—

tan(A±B) = (tanA±tanB)/(1∓tanA tanB)—

sinA+sinB = 2sin((A+B)/2)cos((A-B)/2)—Sum to product

cosA+cosB = 2cos((A+B)/2)cos((A-B)/2)—Sum to product

cosA-cosB = -2sin((A+B)/2)sin((A-B)/2)—Difference to product

Laws & Area

a/sin A = b/sin B = c/sin C—Law of Sines

c² = a² + b² - 2ab cos C—Law of Cosines

Area = ½ab sin C—Two sides + included angle

Area = √(s(s-a)(s-b)(s-c))—Heron's formula, s=(a+b+c)/2

tan A = a sin B/(b - a cos B)—Law of Tangents variant

Inverse Trig & Graphs

arcsin: domain [-1,1], range [-π/2,π/2]—

arccos: domain [-1,1], range [0,π]—

arctan: domain ℝ, range (-π/2,π/2)—

y = A sin(Bx+C)+D—General sinusoidal

Amplitude = |A|—

Period = 2π/|B|—

Phase shift = -C/B—

Vertical shift = D—

allprintabledoc.com