section14

COSINE OF AN ANGLE, THE PYTHAGOREAN THEOREM.

Similarly, the cosine of the angle B is

Theorem 14.1 Cosine of an angle depends only on the degree measure of the angle and does not depend on the size of a triangle that contains the angle.

Proof Consider two unequal right triangles ABC and DEF. Suppose that

Then we need to show that cos A = cos D or, equivalently, AC/AB = DF/DE.

Let's construct a triangle AGH equal to the triangle DEF (we cal always do this according to the axiom 3).

By the definition of the cosine of the angle A we have that

But AH = DF and AG = DE. Therefore

Theorem 14.2 In a right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse then

Proof Consider a right triangle ABC with the right angle C. We need to show that

Let's construct the altitude CD. For the right triangles ABC and ADC cos A = AC/AB = AD/AC.

So,

Similarly, for the right triangles ABC and DBC cos B = BC/AB = DB/CB.

Finally,

Corollary 1 In a right triangle the length of any leg is less than the length of a hypotenuse.

EXERCISE SET #14

Problem 1. Suppose that the length of a diagonal of a square is equal to inches. Find the length of a side of the square.

Problem 2. Is it possible for a right triangle to have sides with lengths equal to 5, 6 and 7 inches? Why?