The document discusses the law of tangents, which sets up a ratio relationship between the sum and difference of two sides of a triangle and the tangents of half the sum and differences of the angles opposite the sides. An example problem demonstrates using the law of tangents to solve for a missing side of a triangle given two sides and one angle measure. The full working of the problem is shown, applying the tangent law formulas and then using the law of sines to find the final missing side measurement.

1. Law of Tangent

2. • 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?

3. 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)]

4. 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.

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

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

7. Determine A + B.
A + B = 180° -
A + B = 180 – 80 = 100.
Solution:

8. Solution:
Implies:

9. Therefore:
Solution:

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

11. Since, A = 69.5,
B = 100 - 69.5 = 30.5.
So, we get
A = 70 and B = 30.
We have the values of A an B so we will start solving
the equation for side c by using the law of sine.
Solution:

12. Side c = 54 units.
Solution:

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