In our exercise we are asked to evaluate the cosine of the smaller acute angle of a triangle with sides lengths of 10, 24 and 26 centimeters. First, we will determine if this triangle is acute, obtuse or right. To do this, we will use the Converse of the Pythagorean Theorem.
Since the sum of the squares of the two shortest sides is equal to the square of the longest side, this triangle is right. Now let's recall the definition of the cosine of an angle.
If△ ABCis a right triangle with acute∠ A,
then the cosine of∠ Ais the ratio of the length of the leg adjacent∠ A to the length of the hypotenuse.
Having this definition in mind, we will make a picture and name the vertices. The right angle will be located opposite the longest side.
Notice that in this triangle the smaller acute angle lies opposite the shorter leg. Therefore, we need to find the cosine of ∠ A. Now we will write an appropriate ratio using the definition we recalled.
cos A=24/26≈ 0.92
The cosine of the smaller acute angle in this triangle is approximately 0.92.
