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Trigonometry formula solvers

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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.

Arc Length

s=rθs = r \theta

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

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.

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.

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.

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.

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.

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.

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 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 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.

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 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.

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.

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.

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.

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 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 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.

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.

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.