Trigonometry & Geometry Reference | FocusMath

Formula sheet

15 formulas

Pythagorean theorem

a^{2} + b^{2} = c^{2}

Variables: $c$ : hypotenuse; $a, b$ : legs
Remember: Applies only to right triangles.

Sine ratio

sin θ = \frac{opposite}{hypotenuse}

Variables: $θ$ : acute angle
Remember: SOH

Cosine ratio

cos θ = \frac{adjacent}{hypotenuse}

Variables: $θ$ : acute angle
Remember: CAH

Tangent ratio

tan θ = \frac{opposite}{adjacent}

Variables: $θ$ : acute angle
Remember: TOA

Reciprocal identities

csc x = \frac{1}{sin x}, sec x = \frac{1}{cos x}, cot x = \frac{1}{tan x}

Variables: denominators must be nonzero
Remember: Each reciprocal pairs with its primary ratio.

Quotient identities

tan x = \frac{sin x}{cos x}, cot x = \frac{cos x}{sin x}

Variables: denominators must be nonzero
Remember: Write tan or cot using sine and cosine.

Pythagorean identities

sin^{2} x + cos^{2} x = 1

Variables: $x$ : angle
Remember: Divide by $sin^{2} x$ or $cos^{2} x$ to derive the other forms.

Compound angles

sin (A \pm B) = sin A cos B \pm cos A sin B

Variables: $A, B$ : angles
Remember: The sine sign is preserved.

Cosine compound angles

cos (A \pm B) = cos A cos B \mp sin A sin B

Variables: $A, B$ : angles
Remember: The cosine sign reverses.

Degree–radian conversion

18 0^{\circ} = π rad

Variables: angle measure
Remember: Multiply degrees by $\frac{π}{180}$ or radians by $\frac{180}{π}$ .

Circle

C = 2 π r, A = π r^{2}

Variables: $r$ : radius; $C$ : circumference
Remember: Use consistent length units.

Prism and cylinder volume

V = A_{ba se} h

Variables: $h$ : perpendicular height
Remember: Cylinder base area is $π r^{2}$ .

Pyramid and cone volume

V = \frac{1}{3} A_{ba se} h

Variables: $h$ : perpendicular height
Remember: One third of the matching prism.

Sphere

S A = 4 π r^{2}, V = \frac{4}{3} π r^{3}

Variables: $r$ : radius
Remember: Area uses square units; volume uses cubic units.

Similarity

\frac{L _{2}}{L _{1}} = k, \frac{A _{2}}{A _{1}} = k^{2}, \frac{V _{2}}{V _{1}} = k^{3}

Variables: $k$ : linear scale factor
Remember: Dimensions determine the scale-factor exponent.

Worked examples

Find a missing leg

A right triangle has hypotenuse $13$ and one leg $5$ .

Show solution

Let the missing leg be $b$ .
Use $5^{2} + b^{2} = 1 3^{2}$ .
$b^{2} = 169 - 25 = 144$ .
$b = 144 = 12$ units.

Variable glossary

$θ$: angle
$r$: radius in length units
$h$: perpendicular height
$k$: similarity scale factor

Student tools

Open graph studio Guided calculators