**30% De 60** – We have seen that we can calculate trigonometric ratios using a calculator or geometric methods. The calculator finds the ratio in rounded decimal form, the geometric method is not very comfortable.

There are some trigonometric ratios of angles that are easy to state accurately, and because of their widespread use, these should be kept in mind.

## 30% De 60

To find the reason for the 45° angle, we’ll start with the right triangle it appears in. We know that the sum of the two acute angles of a right triangle is 90°. Therefore, if a right triangle has an angle of 45°, it must have two angles. If a triangle has two equal angles, the sides opposite those angles must also be equal. So we’ll start with an isosceles right triangle, and to simplify the calculations we can choose a triangle with side length 1.

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Use the navigation bar of the applet below to view the construction of the triangle and the calculation of the ratio step by step.

To find the ratio of the 60° angle, we must start with a right triangle whose acute angle is this measure. We can easily remember such triangles. We start with an equilateral triangle, which we know has three 60° angles. We plot the height and consider one of the two right triangles it is divided into. Since the height and the bisector of an equilateral triangle are the same, the height falls at the midpoint of the side, so one leg measures exactly half the hypotenuse. Since the height also coincides with the bisector, the angle will also be bisected. So we have a right triangle with one angle of 60 degrees and another angle of 30 degrees, this will help us to find the trigonometric ratio of the two angles. To simplify the calculations, we take 2 units as the diagonal, so the leg next to the 60° angle has a measurement of 1.

To find the trigonometric ratios for the 30º angle, we’ll start with the same right triangle that was used for the 60º angle. Remember that the leg opposite the 60 degree angle is the side of the 30 degree angle and the leg on the side of the 60 degree angle is the opposite side of the 30 degree angle. The hypotenuses of both corners are the same. With this, all that remains is to do the calculations and the corresponding simplifications.

Use the navigation bar of the applet below to see the step-by-step scale calculation.

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