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Trigonometry formula sheet

26 formulas · what each one means · a free lesson for each

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steplyclass.com/formula-sheets/trigonometry/ · free, step-by-step lessons for every formula

Angles and radians

Degrees and radians
π radians = 180°
Degrees to radians: × π/180. Radians to degrees: × 180/π.Lesson: Degrees and radians →
Arc length
s = rθ
θ in radians.Lesson: Arc length and sector area →
Sector area
A = 12r2 θ
θ in radians.Lesson: Arc length and sector area →

Trig ratios and the unit circle

SOH CAH TOA
sin θ = opphyp cos θ = adjhyp tan θ = oppadj
Lesson: Right-triangle trigonometry →
Unit circle
(x, y) = (cos θ, sin θ)
A point on the circle of radius 1.Lesson: The unit circle →
Special angles
sin 30° = 12 sin 45° = √22 sin 60° = √32
cos goes the other way: cos 30° = √3/2, cos 60° = 1/2.Lesson: The unit circle →
Signs by quadrant
I: all + · II: sin + · III: tan + · IV: cos +
Lesson: Trig of any angle →
Graphs
y = A sin(B(x − C)) + D
Amplitude |A|, period 2π/|B|, shifted C right and D up.Lesson: Graphs of sine and cosine →
Period of tangent
π|B|
Lesson: Graph of tangent →
Inverse trig ranges
sin−1: [−90°, 90°] cos−1: [0°, 180°] tan−1: (−90°, 90°)
Each gives one angle.Lesson: Inverse trig values →

Basic identities

Reciprocal
csc θ = 1sin θ sec θ = 1cos θ cot θ = 1tan θ
Lesson: Trig identities →
Quotient
tan θ = sin θcos θ cot θ = cos θsin θ
Lesson: Trig identities →
Pythagorean
sin2 θ + cos2 θ = 1
Pythagoras on the unit circle.Lesson: Trig identities →
Pythagorean (tan)
1 + tan2 θ = sec2 θ
Divide the first one by cos²θ.Lesson: Trig identities →
Pythagorean (cot)
1 + cot2 θ = csc2 θ
Divide the first one by sin²θ.Lesson: Trig identities →
Even and odd
cos(−θ) = cos θ sin(−θ) = −sin θ tan(−θ) = −tan θ
Lesson: Trig identities →
Cofunction
sin θ = cos(90° − θ) tan θ = cot(90° − θ)
Lesson: sin of one angle = cos of the other →

Sum and difference identities

Sine
sin(A ± B) = sin A cos B ± cos A sin B
Lesson: Sum and difference identities →
Cosine
cos(A ± B) = cos A cos B ∓ sin A sin B
The sign turns over.Lesson: Sum and difference identities →
Tangent
tan(A ± B) = tan A ± tan B1 ∓ tan A tan B
Lesson: Sum and difference identities →

Double-angle identities

Sine
sin 2θ = 2 sin θ cos θ
Lesson: Double-angle identities →
Cosine
cos 2θ = cos2 θ − sin2 θ = 2cos2 θ − 1 = 1 − 2sin2 θ
Three forms: pick the one with what you know.Lesson: Double-angle identities →
Tangent
tan 2θ = 2 tan θ1 − tan2 θ
Lesson: Double-angle identities →

Any triangle

Law of sines
asin A = bsin B = csin C
Use with a side and the angle across from it.Lesson: The law of sines →
Law of cosines
c2 = a2 + b2 − 2ab cos C
Use with three sides, or two sides and the angle between.Lesson: The law of cosines →
Area with sine
A = 12ab sin C
Two sides and the angle between them.Lesson: Area of a triangle with sine →

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