True or false? If A and B are angles of a triangle such that A > B, then cos A > cos B.
It is false. The smaller angle can have a greater value.
To determine whether the statement is true or false, we need to understand the relationship between the measures of angles and their corresponding cosine values in a triangle.
In a triangle, the cosine of an angle is equal to the ratio of the length of the side adjacent to that angle to the hypotenuse. Here, we consider a right triangle (since the cosine function assumes value for right angles) to explain this concept.
Let's assume A and B are angles of the triangle, with A being the larger angle (A > B). Assuming that A and B are both acute angles, we can create a right triangle with A and B as two of its angles.
In this right triangle, the cosine of angle A is given by the ratio of the adjacent side to the hypotenuse, expressed as cos A = adjacent/hypotenuse. Similarly, the cosine of angle B is given by the ratio of the adjacent side to the hypotenuse, expressed as cos B = adjacent/hypotenuse.
Since angle A is larger than angle B, we can conclude that the side adjacent to angle A is longer than the side adjacent to angle B. As a result, cos A = adjacent/hypotenuse will have a larger value than cos B = adjacent/hypotenuse.
Therefore, the statement "If A and B are angles of a triangle such that A > B, then cos A > cos B" is true.