Law of Tangent

• Law of tangents sets a ratio relationship between the sum and difference of two sides of a triangle, to the tangents of half the sum and differences of the angles opposite to the sides.

• The tangent law is applied and calculated the way law of sine and cosine are applied to identify the measures of a given triangle.

What is law of tangents?

Law of sines formulas

a-b = tan [1/2(A-B)]a+b tan [1/2(A+B)]

 b-c = tan [1/2(B-C)]b+c tan [1/2(B+C)]

c-a = tan [1/2(C-A)]c+a tan [1/2(C+A)]

Example

Use the law of tangents to solve the following triangle:

a is 52, b is 28, and angle   = 80 degrees. Two sides and one angle of the triangle are given.

Let us start by filling the given values into the tangent law formula and simplify it.

Solution:

We will now multiply each side by the denominator on the right.

Solution:

Determine A + B.

A + B = 180° - A + B = 180 – 80 = 100.

Solution:

Solution:

Implies:

Therefore:

Solution:

Now, let us solve the system of equations for A + B and A – B.

Solution:

Since, A = 69.5, B = 100 - 69.5 = 30.5.So, we getA = 70 and B = 30.

We have the values of A an B so we will start solvingthe equation for side c by using the law of sine.

Solution:

Side c = 54 units.

Solution:

Still got

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