Скачать или смотреть Trigonometric Identities & Proofs Pythagorean Identities, Sum & Difference Formulas

Trigonometric Identities & Proofs Pythagorean Identities, Sum & Difference Formulas.

This is a song about trigonometry identities and how to prove them.
Clear steps, clear sound, one topic at a time.

On the unit circle, radius equals one.
Horizontal is cosine, vertical is sine.
By the Pythagorean Theorem, squares add to one.
So: sine squared plus cosine squared equals one.
From that single fact, more relations arrive.
Divide by cosine squared to keep it alive:
One plus tangent squared equals secant squared.
Or divide by sine squared if that helps more:
One plus cotangent squared equals cosecant squared.

Circle gives the coordinates, theorem gives the sum.
Rename x as cosine, rename y as sine, done.

Sine squared plus cosine squared equals one.
That identity powers almost every run.
When problems look heavy and you need a plan,
Check the Pythagorean, yes you can.

Angles can combine; we name them alpha and beta.
Addition or subtraction, we keep the pattern straight.
For sine of a sum, the sign stays the same:
Sine alpha times cosine beta,
plus cosine alpha times sine beta.
For sine of a difference, the sign follows through:
Sine alpha times cosine beta,
minus cosine alpha times sine beta.
For cosine of a sum, the sign switches lane:
Cosine alpha times cosine beta,
minus sine alpha times sine beta.
For cosine of a difference, flip that sign again:
Cosine alpha times cosine beta,
plus sine alpha times sine beta.
For tangent, use sine over cosine and divide both lines:
Tangent of a sum equals tangent alpha plus tangent beta
over one minus tangent alpha times tangent beta.
For a difference, the numerator uses a minus,
and the denominator uses a plus.

Sine keeps the sign.
Cosine switches sign.
Tangent is a fraction with an extra line.

Start with a unit vector pointing along the x axis.
Rotate by alpha; coordinates are cosine alpha, sine alpha.
Rotate again by beta; total turn is alpha plus beta.
Rotation changes x by cosine times x minus sine times y.
Rotation changes y by sine times x plus cosine times y.
Plug the first coordinates into these two updates.
The new x equals cosine of the sum.
The new y equals sine of the sum.
Collect the terms and read them out loud:
Cosine of a sum becomes cosine alpha cosine beta
minus sine alpha sine beta.
Sine of a sum becomes sine alpha cosine beta
plus cosine alpha sine beta.
For differences, replace beta by negative beta
and use that cosine is even and sine is odd.
That closes the proof.

Set the two angles equal; let beta match alpha.
Then the sum becomes a double angle.
Sine of double alpha equals two times
sine alpha times cosine alpha.
Cosine of double alpha has three equal forms:
Cosine squared alpha minus sine squared alpha,
or one minus two times sine squared alpha,
or two times cosine squared alpha minus one.
Tangent of double alpha equals
two times tangent alpha divided by
one minus tangent squared alpha.

Start with cosine squared minus sine squared.
Replace sine squared by one minus cosine squared.
That gives two times cosine squared minus one.
Or replace cosine squared by one minus sine squared.
That gives one minus two times sine squared.
All three forms are the same; choose the one that fits the problem.

Sine squared plus cosine squared equals one.
Build every step from that and you have won.
Sums, differences, doubles in a clean design:
Keep the patterns straight and you will be fine.

Start from the circle to get the Pythagorean line.
Sine keeps the sign; cosine switches sign.
For tangent, write a fraction with a one on the base.
For double angles, set the two the same and replace.
Sing it while you practice; check each step you take.
Accurate moves make every problem break.