Derive the Trigonometric Identities (Unit Circle)
This examples demonstrates how to derive the trigonometric identities using the trigonometric functions and the geometry of the unit circle. This examples contains how to:
- Derive Pythagorean Identity (Unit Circle)
- Derive Sum of Two Angles Identities (Unit Circle)
- Derive Difference of Two Angles Identities (Unit Circle)
- Derive Double Angle Identities (Unit Circle)
- Derive Half Angle Identities (Unit Circle)
The pythagorean identity relates the squared sides of the right triangle on the unit circle together. However, it can be hard to see how the squared sides translates to the geometry of the right triangle on the unit circle. This example shows the steps to finding the relationship between the length of and the squared length of the and .
Start with the right-triangle on the unit circle as shown below defined by the point . Label the lengths of the adjacent and opposite sides in terms of the angle of the triangle.
The lengths of the adjacent and opposite side can be solved for by applying the definitions of the sine and cosine functions.
Next, divide the right triangle into two similar triangles by drawing a line from the corner of the right-triangle perpendicular to its hypotenuse. This is shown below:
Then find the length of the adjacent side, labeled with the variable , of the first similar triangle shown below:
Apply the definition of cosine and then substitute the length of in for the hypotenuse and the length of for the adjacent side and then solve for .
Repeat this process to find the length of the opposite side of the second similar triangle, labeled with the variable :
Apply the definition of sine and then substitute the length of in for the hypotenuse and the length of for the opposite side and then solve for .
Finally, we can observe that the hypotenuse the right triangle of lenght can be expressed as the sum of the lengths and which gives us pythagorean’s identity:
This example shows how to derive the sum of two angle identities, shown below, using the geometry of the unit circle. This example is part of a series of examples the show how to derive the trigonometric identities using the unit circle.
Start by drawing the cosine and sine lengths associated with the sum of two angles.
Figure 1: Unit CircleTest figure.
- Step number two.
- Step number three.
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The trigonometric identites are a set of equations derived from the properties of the right triangle and the circle.
There are six trigonometric functions that relate to the geometry of the right-triangle sine, cosine, tangent, cosecant, secant, and cotangent. The functions take the angle of a right triangle as input and return a ratio of two of its sides.
The unit circle is a circle of radius one placed at the origin of the coordinate system. This article discusses how the unit circle represents the output of the trigonometric functions for all real numbers.
To geometrically derive the pythagorean identity, divide the right triangle into two similar triangles by drawing a line from the right angle corner of the right-triangle perpendicular to its hypotenuse. Then solve for each triangle's length on the hypotenuse.
The sum of two angles addition formula can be derived using a quadrilateral inscribed on a circle of diameter 1.
