Sine and Cosine Angles Sum and Difference Identities | self-made

Sine and Cosine Angles Sum and Difference Identities

Derivated using a right-angled triangle inscribed into a rectangle.

To show,

The sum and difference for sine and cosine functions | self-made

it is sufficient to make one drawing and note essentials.

What you should already know?

• the relationship between sides of right-angled triangles,
• that cosine is an even and the sine is an odd function.

The drawing

Start by inscribing a right-angled triangle with a unit-length hypotenuse into a rectangle

Inscribe a right-angled triangle in a rectangle to have a unit hypotenuse | self-made

and use relations for cosine and sine to evaluate its sides.

Measure its sides using sine and cosine identities in right-angled triangles | self-made

Then, evaluate the below right-angled triangle’s sides. Note, its hypotenuse equals cosβ.

Measure sides of the below right-angled triangle | self-made

Once done, on the triangle adjacent to the right side of the inscribed triangle, calculate its undermost angle. Note, it is part of a straight angle. Hence, ∠= 180°-90°-(90°-α)=α.

Identify the angle in the right-angled triangle adjacent to the right side of the inscribed one | self-made

Evaluate its sides.

Measure the sides of the adjacent right-angled triangle | self-made

Repeating the very same reasoning to the triangle adjacent to the hypotenuse of the inscribed triangle, its rightmost angle is equal to α+β.