What is a trigonometric circle?

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It is a diagram planned to make easier the explanation about the existing relations among the several trigonometric ratio. We draw a circumference with a radius 1 and two orthogonal axes (perpendicular) which go through the centre. In order to use the trigonometric circle, we agree that the origin side of every angle is the semi-positive x-axis.

trigonometric circle

We can see through the above image that as the circle has radius 1, which is coloured green, the `sin alpha` can be measured in the y-axis. It happens because there is a formula that says: `sin alpha = "opposite cathetus" / "hypotenuse"` and as the hypotenuse corresponds to the circumference radius, and so it measures 1, the formula can be simply the following: `sin alpha = "opposite cathetus"` .

In the circumference we can also find the measures for the cosine by using a similar reasoning. In the rectangle triangle, the `cos alpha` can be measured through the x-axis. The formula tells us that `cos alpha = "adjacent cathetus" / "hypotenuse"`, and once more, as the hypotenuse measures 1, we can simply have `cos alpha = "adjacent cathetus"` .

Is the trigonometric circle useful for anything else?

Undoubtedly it is! Besides allowing us to easily identify the Sino, the cosine and the tangent, it is also useful for other circumstances such as to check what is the quadrant a certain angle belongs to; to reduce angles to the first quadrant; to check the signal of an angle trigonometric ratios; to compare the order of magnitude between two trigonometric ratios, etc.

What are notable angles?

One of the greatest advantages concerning the use of the trigonometric circle was the representation of trigonometric ratios of every angle, no matter if it is positive or negative, acute or obtuse, and even the representation of angles measuring more than 360º. However, there are some angles whose amplitudes have a peculiar importance. That’s the case of all the axes, that is, the multiples of 90º and also three very important angles called notable angles: 30º, 45º and 60º.

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