Precalculus

Trigonometry

Trigonometry Logo

Law of Cosines

Solved Problems

Example 1.

Find the least diagonal of a regular hexagon with side length a.

Solution.

A regular hexagon with the least diagonal
Figure 4.

The internal angle in a regular hexagon is 120. Using the Cosine Rule, we get

d2=a2+a22a2cos120=2a22a2(12)=2a2+a2=3a2.

Then the least diagonal is equal to

d=a3.

Example 2.

Two sides of a triangle are equal to a and b. The angle between them is 135. Find the third side c.

Solution.

By the Cosine Rule, we have

c2=a2+b22abcos135.

Calculate cos135 using the Reduction Identity:

cos135=cos(18045)=cos45=22.

Hence,

c2=a2+b22ab(22)=a2+b2+ab2,c=a2+b2+ab2.

Example 3.

The middle side of a triangle is 1 greater than the least side and 1 less than the greatest side. The cosine of the middle angle is equal to 35. Find the area of the triangle.

Solution.

Let be the length of the middle side. Then the other sides of the triangles are equal to and

In any triangle, the mid-sized side and mid-sized angle are opposite to each other. Hence, using the Law fo Cosines, we can write:

Solve this equation for

The negative root does not make sense. Hence, The other two sides are equal and

Note that this triangle satisfies the Pythagorean theorem:

In such a case, the area of the triangle is

Example 4.

In a triangle the side is the midline parallel to the side is and the angle Find the side

Solution.

A triangle with a midline
Figure 5.

The midline is half the length of the side that is,

Let be the length of the side Write the Cosine Rule for the triangle

Recall that Then

Find the discriminant of the quadratic equation:

The roots of the equation are

We choose the positive root, so

Example 5.

Given a triangle whose sides are and is the length of the median drawn from the vertex to side Prove that

Solution.

A triangle with the median from vertex A to side a.
Figure 6.

Use the Law of Cosines for the triangle

Solve this equation for

Apply now the Law of Cosines to the triangle taking into account that This yields:

Subctitute found above:

Example 6.

Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Solution.

Consider a parallelogram with sides and diagonals

A parallelogram with the diagonals d1 and d2
Figure 7.

Denote the angle by The other angle is then equal to

Applying the Law of Cosines to the triangle we can write:

By the Reduction Identity,

This yields:

Similarly we find the square of the diagonal

or

Calculate the sum of the squares of the diagonals:

The right-hand side represents the sum of the squares of four sides of the parallelogram.

Page 1 Page 2