Question

Finding an Angle Two sides of a triangle have lengths \(a\) and \(b,\) and the angle between them is \(\theta\) . What value of \(\theta\) will maximize the triangle's area? [Hint: \(A=(1 / 2) a b \sin \theta . ]\)

Short answer

The angle \( \theta \) that will maximize the triangle's area is 90 degrees.

Step-by-step solution

Write down the area equation for the triangle

The area of the triangle is given as \(A=(1 / 2) a b \sin \theta \).

Realize the range of possible values for \(\sin \theta\)

The range of sine function is from -1 to +1. However, since we are dealing with angles in a triangle, the acceptable range for \(\sin \theta\) is [0, 1]. That is because the angle \(\theta\) of a triangle can vary from 0 to 180 degrees, but for positive areas we should only consider 0 to 90 degrees.

Find the maximum value for \(\sin \theta\)

To maximize the area of the triangle, we want to maximize the value of \(\sin \theta\) because both a and b are fixed lengths. The maximum value of \(\sin \theta\) on the interval [0, 90 degrees] is 1.

Find the corresponding angle \(\theta\)

We then determine what angle \(\theta\) gives us the maximum value for \(\sin \theta\). Looking at the sine function we find that \(\sin \theta = 1\) at \(\theta =90^{\circ}\).