show that,t for any triangle abc, even if b or c is an obtuse angle, a = b cos c + c cos b. use the law of sines to deduce the "addition formula" sin (b + c) = sin b cos c + sin c cos b.

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19-07-2022
Mathematics

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show that,t for any triangle abc, even if b or c is an obtuse angle, a = b cos c + c cos b. use the law of sines to deduce the "addition formula" sin (b + c) = sin b cos c + sin c cos b.

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pranavgera011 pranavgera011 · Answer 1 · 2022-07-27T06:27:13+02:00

For the triangle ABC, if B or C is an obtuse angle, then a = b cos C + c cos B according to the law of sines and the addition formula.

What is the law of sines states?

For a triangle ABC with angles ∠A, ∠B, and ∠C and with sides a, b, and c

Thus, we can write,

sinA/a = sinB/b = sinC/c

or

a/sinA = b/sinB = c/sinC

Calculation:

It is given that in a ΔABC,

The ∠B or ∠C is an obtuse angle.

So, ∠A = 180° - (B + C)

⇒ sin A = sin (180° - (B + C))

⇒ sin A = sin (B + C) (since we know that sin(180° - θ) = sinθ)

From the addition formula,

sin (B + C) = sin B cos C + sin C cos B

⇒ sin A = sin (B + C)

⇒ sin A = sin B cos C + sin C cos B

From the law of sines,

a/sinA = b/sinB = c/sinC = K

⇒ aK =sin A, bK = sin B, cK = sin C

On substituting,

⇒ sin A = sin B cos C + sin C cos B

⇒ aK = bK cos C + cK cos B

⇒ K(a) = K(b cos C + c cos B)

∴ a = b cos C + c cos B

Hence it is proved.

Learn more about the law of sines here:

https://brainly.com/question/2807639

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