Angle Sum and Difference Identities

Let $△ A F D$ be a right triangle with hypotenuse $1$ and an acute angle with measure $x + y .$

By definition, the sine of an angle is the ratio between the lengths of the opposite side and the hypotenuse.

sin (x + y) = \frac{D F}{1} ⇓ D F = s i n (x + y)

The idea now is to rewrite

D F

in terms of

sin x,

sin y,

cos x,

and

cos y .

To do it, draw a ray so that

∠ A

is divided into two angles with measures

x

and

y .

Let

C

be a point on this ray such that

△ A C D

and

△ A B C

are right triangles.

Consider

△ A C D .

By calculating the sine and cosine of

x,

the legs of this triangle can be rewritten.

sin x = \frac{D C}{1} \Rightarrow D C = sin x cos x = \frac{A C}{1} \Rightarrow A C = cos x

Now consider

△ A B C .

Knowing that

A C = cos x,

the sine of

y

can be used to write

B C

in terms of

x

and

y .

sin y = \frac{B C}{A C}

Solve for

B C

Substitute

$A C = c o s x$

sin y = \frac{B C}{c o s x}

MultEqn

$LHS \cdot cos x = RHS \cdot cos x$

cos x sin y = B C

RearrangeEqn

Rearrange equation

B C = cos x sin y

Let

G

be the point of intersection between

\overline{F D}

and

\overline{A C} .

Notice that

∠ A G F ≅ ∠ D G C

by the Vertical Angles Theorem.

By the Third Angle Theorem, it is known that $∠ G A F ≅ ∠ G D C .$ Therefore, $m ∠ G D C = y .$

Since the purpose is to rewrite $D F,$ plot a point $E$ on $\overline{D F}$ such that $\overline{E C} ∥ \overline{A B} .$ This way a rectangle $E C B F$ is formed. The opposite sides of a rectangle have the same length, so $E F$ and $C B$ are equal. Also, $\overline{C E} ⊥ \overline{D F}$ makes $△ C E D$ a right triangle.

Consequently,

E F = c o s x s i n y

and

D E

can be written in terms of

sin x

and

cos y

using the cosine ratio.

cos y = \frac{D E}{sin x} ⇓ D E = s i n x c o s y

Finally, by the Segment Addition Postulate,

D F

is equal to the sum of

D E

and

E F .

All these lengths have been rewritten in terms of the sine and cosine of

x

and

y .

D F = D E + E F ⇓ s i n (x + y) = s i n x c o s y + c o s x s i n y

This concludes the proof of the first identity. The other identities can be proven using similar reasoning.

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Rule

Angle Sum and Difference Identities

Proof

Extra

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