easy with sin(a+b) proof : vuja-de

easy with sin(A+B) proof

Some simple concepts in maths and science are presented in disconnected pieces and make it hard for students to understand. The same can be explained in a refreshing new and well connected form. This “vuja-de” series brings out such refreshing new views from nubtrek.com

trigonometric ratios of compound angles

Students learns first trigonometric ratios for right-triangles and then they move on to learn trigonometric ratios in unit circle form. The unit circle form extends the trigonometric ratios to angles beyond 90°.

Later they learn the trigonometric identities of compound angles, like sin(A+B).

sin(A+B) identity with triangles

When starting on trigonometric ratios of compound angles, a proof with triangles is presented. Students learn the identity sin(A+B) = sin A cos B + cos A sin B with triangles.

proof in triangular form

Students are left wondering what if one or more angle is greater than 90°?

Reference: Khan Academy

sin(A+B) proof in unit-circle form from nubtrek

nubtrek extends the proof for sin(A+B) in unit circle — in a refreshing new and well connected form.

• unit circle definition of trigonometric ratios is shown for angles A, B, and A+B

unit circle definition of trigonometric ratios for A, B, A+B

• Equating length of line segments PQ1 and RT, it is proven that sin(A+B) = sin A cos B + cos A sin B.

Proof of sin(A+B) identity

• Q1 is (sinB, cosB)
• PQ1 = 2–2sinAcosB-2cosAsinB
• RT = 2–2sin(A+B)
• PQ1 = RT proves the required identity.

This proof in nubtrek holds for angle greater than 90° and is in accordance with the unit circle form of trigonometric ratios.

Reference: Trigonometric Ratios of Compound Angles in nubtrek

other identities

• For cos(A+B), sin(A-B) and cos(A-B), the proven identity sin(A+B) is used as given below.
• cos(A+B) = sin(90-A-B)
• sin(A-B) = sin(A+(-B))
• cos(A-B) = cos(A+(-B))
• Though the proofs of cos(A+B), sin(A-B) and cos(A-B) based on the proven identity sin(A+B) works well, as an added information, the following geometrical proofs are presented for students.
• cos(A+B) identity is proven by equating lengths of line segments RS and PQ2

Proof of cos(A+B) identity

• sin(A-B) identity is proven by equating lengths of line segments RT and PQ1
• cos(A-B) identity is proven by equating lengths of line segments RS and PQ

Proofs for sin(A-B) and cos(A-B) identities

Thanks for reading.