# Two equations in two unknowns with trig functions

#### oldcodger

##### New member
These two equations are needed to find the solution for a land surveying problem. The equations represent the summations of the latitudes and departures for a closed traverse wherein one leg of the traverse has the length known but the bearing is unknown and another leg where the bearing is known but the length is unknown. I solved this problem about 15 years ago but my algebra fails me now.
L cos a + R cos B + D = 0
L sin a + R sin B + C = 0
I need to solve for L (distance) and B (angle).

I am not a student needing a problem solved. This problem represents a needed solution for laying out an intersection on a highway using old school methods.

#### LCKurtz

##### Junior Member
Do you have values for R, C, and D? If so, would a numerical solution help you? Or are you hoping for a nice simple formula? Not being familiar with surveying terms, I wouldn't mind seeing a picture of the situation. I could crunch some numbers in Maple for you.

#### Dr.Peterson

##### Elite Member
I assume a, C, D, and R are considered to be known. I would solve each equation for L and set those equal, then solve for B. Then plug that into either equation to find L.

The trig equation is a little tricky; one approach is to rewrite a sum of multiples of sine and cosine as a sine or cosine of a sum or difference.

• topsquark

#### oldcodger

##### New member
Sorry for my late response I have been away from home for a few days. I finally got the equation I needed. Attached is a sketch showing the layout and a sheet showing the equation. I proceeded with Dr.Petersons suggestion for solving each equation for L and setting them equal and solve for sin B. As you can see the solution is a quadratic equation caused by the a trig substitution. Like I said in my post presenting the problem my algebra just failed me until I had some help from this site. Thanks everybody.

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