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Trigonometry formula solvers

Circular Sector Area

A=12r2θA = \frac{1}{2} r^{2} \theta

GeometryTrigonometryArea of a pie-slice sector of a circle from its radius and central angle in radians.

Arc Length

s=rθs = r \theta

GeometryTrigonometryLength of a circular arc as radius times central angle in radians.

Right-Triangle Sine Ratio (SOH)

sin⁡θ=oh\sin\theta = \frac{o}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its opposite side and the hypotenuse.

Right-Triangle Cosine Ratio (CAH)

cos⁡θ=ah\cos\theta = \frac{a}{h}

TrigonometryGeometryRelates an acute angle of a right triangle to its adjacent side and the hypotenuse.

Right-Triangle Tangent Ratio (TOA)

tan⁡θ=oa\tan\theta = \frac{o}{a}

TrigonometryGeometryRelates an acute angle of a right triangle to the ratio of its opposite and adjacent legs.

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

TrigonometryGeometryFinds the third side of any triangle from two sides and their included angle, or the angle from all three sides.

Law of Sines

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

TrigonometryGeometryIn any triangle, each side is proportional to the sine of its opposite angle.

Triangle Area (Two Sides and Included Angle)

A=12 absin⁡CA = \tfrac{1}{2}\,ab\sin C

TrigonometryGeometryComputes a triangle's area from two sides and the angle between them, with no height needed.

Direction Angle of a 2D Vector

θ=atan2⁡(vy,  vx)\theta = \operatorname{atan2}(v_y,\; v_x)

Vectors & MatricesTrigonometryAlgebraGives the direction a two-dimensional vector points, measured counterclockwise from the positive x-axis, from its two components.

x-Component from Magnitude and Angle

vx=∣v⃗∣cos⁡θv_x = |\vec{v}| \cos\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its horizontal part from the vector's length and the angle it makes with the positive x-axis.

y-Component from Magnitude and Angle

vy=∣v⃗∣sin⁡θv_y = |\vec{v}| \sin\theta

Vectors & MatricesTrigonometryPhysicsResolves a vector into its vertical part from the vector's length and the angle it makes with the positive x-axis.

Dot Product from Magnitudes and Included Angle

a⃗⋅b⃗=∣a⃗∣ ∣b⃗∣cos⁡θ\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta

Vectors & MatricesTrigonometryPhysicsGives the scalar product of two vectors from their lengths and the angle between them, the geometric face of the dot product.

Angle Between Two 2D Vectors (Components)

θ=arccos⁡(axbx+aybyax2+ay2 bx2+by2)\theta = \arccos\left(\frac{a_x b_x + a_y b_y}{\sqrt{a_x^2+a_y^2}\,\sqrt{b_x^2+b_y^2}}\right)

Vectors & MatricesTrigonometryAlgebraFinds the angle separating two plane vectors directly from their four components, by way of the normalized dot product.

Cross Product Magnitude

∣a⃗×b⃗∣=∣a⃗∣ ∣b⃗∣sin⁡θ|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta

Vectors & MatricesTrigonometryPhysicsGives the length of the cross product of two vectors from their magnitudes and the angle between them, equal to the area they span.

Resultant of Two Vectors at an Angle

R=A2+B2+2ABcos⁡θR = \sqrt{A^2 + B^2 + 2AB\cos\theta}

Vectors & MatricesTrigonometryPhysicsFinds the magnitude of the sum of two vectors from their lengths and the angle between them, the parallelogram rule in one equation.

Grade to Slope Angle

θ=arctan⁡ ⁣(G100)\theta = \arctan\!\left(\frac{G}{100}\right)

Civil & SurveyingTrigonometryConverts a percent grade into the slope angle measured from horizontal, and back again with the tangent function.

Horizontal Curve Tangent Length

T=Rtan⁡Δ2T = R\tan\frac{\Delta}{2}

Civil & SurveyingTrigonometryDistance from the point of intersection back to the point of curvature, from the curve radius and its total deflection angle.

Horizontal Curve External Distance

E=R(sec⁡Δ2−1)E = R\left(\sec\frac{\Delta}{2} - 1\right)

Civil & SurveyingTrigonometryClearance from the point of intersection to the midpoint of the curve, the distance a curve cuts back from the corner.

Horizontal Curve Middle Ordinate

M=R(1−cos⁡Δ2)M = R\left(1 - \cos\frac{\Delta}{2}\right)

Civil & SurveyingTrigonometryOffset from the middle of the long chord to the middle of the arc, the number that governs sight distance around obstructions.

Horizontal Curve Long Chord

C=2Rsin⁡Δ2C = 2R\sin\frac{\Delta}{2}

Civil & SurveyingTrigonometryStraight-line distance from the point of curvature to the point of tangency, the chord that spans the entire circular curve.

Back Azimuth

αb=α±180∘\alpha_b = \alpha \pm 180^{\circ}

Civil & SurveyingTrigonometryReverses a direction by adding or subtracting one hundred eighty degrees, keeping the result inside the zero to three sixty range.

Latitude of a Traverse Leg

Lat=Lcos⁡α\text{Lat} = L\cos\alpha

Civil & SurveyingTrigonometryNorth-south component of a traverse course, from the measured slope-corrected distance and the azimuth of the line.

Departure of a Traverse Leg

Dep=Lsin⁡α\text{Dep} = L\sin\alpha

Civil & SurveyingTrigonometryEast-west component of a traverse course, taken from the measured horizontal distance and the azimuth of the line.