Difference between revisions of "Proofs of trig identities"
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<math>1+\cot^2=\csc^2</math> | <math>1+\cot^2=\csc^2</math> | ||
+ | ==Sum== | ||
==Sum== | ==Sum== | ||
<math>\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha</math> | <math>\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha</math> | ||
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<math>\tan(\alpha + \beta)=\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}</math> | <math>\tan(\alpha + \beta)=\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}</math> | ||
+ | |||
+ | <math>\sec(\alpha + \beta)=\frac{\sec \alpha \sec \beta}{1 - \tan \alpha \tan \beta}</math> | ||
+ | |||
+ | <math>\csc(\alpha + \beta)=\frac{\csc \alpha \csc \beta}{\cot \alpha + \cot \beta}</math> | ||
+ | |||
+ | <math>\cot(\alpha + \beta)=\frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}</math> | ||
+ | |||
+ | Memory aids: sin is different, cos is same, tan is sum over one minus product, cot is product minus one over sum, sec is like tan but with product of sec on top, csc is like cot but with product of csc on top. | ||
==Double== | ==Double== | ||
<math>\sin(2n)=2\sin n\cos n</math> | <math>\sin(2n)=2\sin n\cos n</math> |
Revision as of 18:47, 22 August 2024
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Contents
Introduction
and are easy to define. I prefer the unit circle definition as it makes these proofs easier to understand. Next, we define some other functions:
Note: I've omitted because it's unnecessary and might clog things up a little.
With a bit of ingenuity, we can create the following diagram:
It has three main triangles: cos-sin-1, 1-tan-sec, and cot-1-csc.
We can note that the functions are correct by similar triangles.
Symmetric identities
If we draw a few copies of the triangle, we get:
The other three can be derived by taking the reciprocals of these three.
Pythagorean identities
Pythagorean identities are easy and there's no algebra involved. In fact, the name Pythagorean is a giveaway of what we should do!
The proof here is very straightforward. We use the pythagorean theorem on giving us or .
Same story here. Applying pythagorean to gives us or .
Same. Pythagorean on gives or .
Conclusion
Even though with the first one and the definitions, we can make the rest from algebra, having a geometric meaning is nice when we want to know what it actually means.
Angle addition and subtraction
When does sin appear? When does sin appear? In the first triangle, of course. Let's make a diagram!
where
The diagram illustrates the identities nicely.
The diagram shows the height of point is . However, the length of is . To compensate, we must divide by to make it the sine. After some *easy* algebra, we arrive at .
The diagram says that it is , but we need to divide by again. We arrive at .
This time, let's use the tan-1-sec triangle.
Wait, is that just the same diagram? No! the labels have changed!
Note: I did some algebra when noting that sin * sec = tan and cos * sec = 1
Looking at the diagram, the height of the new triangle is , but the width is only , so we arrive at
I dunno why most people never use this, but it's right up there in the diagram.
Hey, if you don't need this, stop reading.
We need the third triangle here. I'm going to do something weird, you'll see why when I complete the diagram (I= afly. This page was all made by afly)
Double angle formulas
This is a breeze. Just sub in for sum:
Variations
Since , we can edit the double angle cosine formula a bit. Here are the three most helpful variants:
We can also solve for other expressions:
Sum to Product to Sum
Our angle addition formulas look nasty. Let's try to cancel something.
Chapter 1
Let's start with the formula . Both terms are symmetrical, let's try to cancel out the first one.
and might give us an idea.
Therefore, .
To put this in a nicer form, where:
Chapter 2
We did all we could. Now let's try doing something to the other formula: . Let's try to cancel the first one first. is a lot easier to cancel than so we have to subtract.
So
Or, (same conversions of )
Chapter 3
Next: . Now let's cancel the second one.
So
Or, (same conversions of )
Bonus: Product identity
This is a special identity. I hope this helps you.
and
There's something we can cancel.
If , then it simplifies to
Notice . If we let :
Halved angles
Starting with the identities from the double section:
We take the square root to obtain:
For tangent:
There are two nice variations to know.
Triple angles and more
Triple sums
Triple angles
Third angles
Let and . We get this depressed cubic:
First, divide both sides by -4 and rearrange: . The discriminant
Then,
The solutions are , , and .
A tiny adjustment gives us the cosine third-angle formulas:
, , and .
For tangent:
, , and
All identities
Definition
Symmetric
Pythagorean
Sum
Sum
Memory aids: sin is different, cos is same, tan is sum over one minus product, cot is product minus one over sum, sec is like tan but with product of sec on top, csc is like cot but with product of csc on top.
Double
Sum Product
Product
Halves
3 Sums
Triple
Thirds