Trigonometric identities

From Applied Science

If you understand how to read the unit circle the trigonometric identities are all consequences of it. There are multiple formulas regarding the product, the quotient, addition and subtraction of angles. We don't need to know all of them because they can all be derived from the unit circle. It's more important to understand the unit circle than to memorize the formulas.

The circle's radius is 1 to make the properties easier to grasp. We could have used a circle of any radius greater than 1 because the proportions of the triangle don't change with a larger radius. In the vertical axis we mark the values for sine, while the values for cosine go in the horizontal axis.

Interactive applets:

In the absence of computers it is possible to construct an interactive unit circle with paper. The most important piece is something rigid that can represent the circle's diameter, placed on top of the circle and rotated. I know that there are creative teachers out there in the world that do it. I happen to have had a teacher that did it in high school.

The first consequence of the unit circle is that every angle can be seen two ways, clockwise or anti-clockwise. For example: 315° is the same as 45 clockwise, which means -45°. If we know the sine and cosine of 45°, then -45° is much easier to remember than 315°.


The second consequence is that there are some interesting symmetries. The unit circle shows that [math]\displaystyle{ \cos(\theta) = \cos(-\theta) }[/math] because any angle between 0° and 90° and its opposite correspond to the same value. See that the base of the triangle is the same for two angles, the positive angle and its negative counterpart. With this we've just shown that cosine is an even function. Sine relates to the height of the triangle and the figure shows that, in absolute value, [math]\displaystyle{ \sin(\theta) = |\sin(-\theta)| }[/math]. With sine the dashed line has a positive value for positive angles, but for the negative angles we are along the negative part of the axis. The length of that line itself is positive, but when we calculate ratios we can't just ignore the sign. With this we've just shown that sine is an odd function.


The third consequence is about reflections. We learn in physics and in geometry too that when a light ray or an object hits a flat surface with an angle in between 0° and 90°, the incidence angle is the same as the reflected angle. For example: if the incidence angle is 60°, the reflected is 180° - 60° = 120°. The unit circle shows that [math]\displaystyle{ \sin(\theta) = \sin(\pi - \theta) }[/math]. Notice that the triangle on the left side is mirrored in comparison to the triangle to the right. Both their heights are marked along the positive side of the vertical axis. Now the cosine is a different matter in this case. The mirrored angle has the cosine marked along the negative side of the horizontal axis. Cosine for angles between 90° and 180° yield negative values. The only difference is the sign, because if we take the absolute value they should be equal to their mirrored counterparts to the right side.

With the unit circle having a radius of 1, the fundamental identity is [math]\displaystyle{ \sin^2 \theta + \cos^2 \theta = 1 }[/math] for any angle. It's Pythagoras and also the euclidean distance.

To prove the formulas for sum and difference of angles we need two things: the fundamental identity and the idea that every angle greater than zero can be divided into two angles. Wikipedia and many other sites and textbooks prove the identity for sum of angles without using the unit circle. I want to use it and there is one textbook that does it. There is one textbook that I have that proves it with pure algebra in two lines, with no graph at all.

We begin by placing two points, Q and P. With the unit circle we know the coordinates of each point and know that the distance between the point and the origin is always equal to the radius of one. The unit circle is a means to understand the trigonometric functions and angles, but the other purpose of it is to convert polar coordinates to cartesian coordinates and vice-versa.


Now we rotate the triangle while keeping the angle the same. This gives us new points with new coordinates. The areas of both triangles, POQ and P'OQ' are equal and so are the distances between PQ and P'Q'. How do we calculate a distance between two points? With Pythagoras.

[math]\displaystyle{ \sqrt{(\cos \alpha - \cos \beta)^2 + (\sin \alpha - \sin \beta)^2} = \sqrt{[\cos(\alpha - \beta) - 1]^2 + [\sin(\alpha - \beta) - 0]^2} }[/math]

[math]\displaystyle{ (\cos \alpha - \cos \beta)^2 + (\sin \alpha - \sin \beta)^2 = [\cos(\alpha - \beta) - 1]^2 + [\sin(\alpha - \beta) - 0]^2 }[/math]

[math]\displaystyle{ \cos^2 \alpha - 2\cos \beta \cos \alpha + \cos^2 \beta + \sin^2 \alpha - 2\sin \beta \sin \alpha + \sin^2 \beta = [\cos(\alpha - \beta) - 1]^2 + [\sin(\alpha - \beta) - 0]^2 }[/math]

[math]\displaystyle{ \cos^2 \alpha - 2\cos \beta \cos \alpha + \cos^2 \beta + \sin^2 \alpha - 2\sin \beta \sin \alpha + \sin^2 \beta = \cos^2(\alpha - \beta) - 2 \cos(\alpha - \beta) + 1 + \sin^2(\alpha - \beta) }[/math]

(there is a fundamental identity at the left side that will cancel out with the 1 to the right)

[math]\displaystyle{ - 2\cos \beta \cos \alpha + \cos^2 \beta - 2\sin \beta \sin \alpha + \sin^2 \beta = \cos^2(\alpha - \beta) - 2 \cos(\alpha - \beta) + \sin^2(\alpha - \beta) }[/math]

[math]\displaystyle{ - 2\cos \beta \cos \alpha - 2\sin \beta \sin \alpha + 1 = \cos^2(\alpha - \beta) - 2 \cos(\alpha - \beta) + \sin^2(\alpha - \beta) }[/math]

(the fundamental identity repeats to the right. Let's cancel again)

[math]\displaystyle{ - 2\cos \beta \cos \alpha - 2\sin \beta \sin \alpha = - 2 \cos(\alpha - \beta) }[/math] (now let's cancel the -2 on both sides and we have proven that the cosine of the difference of angles identity)

[math]\displaystyle{ \cos(\alpha - \beta) = \cos \beta \cos \alpha + \sin \beta \sin \alpha }[/math]

The same reasoning can be repeated to prove the other identities.

Reference: https://opentextbc.ca/algebratrigonometryopenstax/chapter/sum-and-difference-identities/

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