MHB Find Angle A for Triangle: Sin2 B + Sin2 C = Sin B Sin C

[anemone]

For the triangle with angles $A, B, C$, the following trigonometric equality holds.

$$\sin^2 B +\sin^2 C-\sin^2 A=\sin B \sin C$$

Find the measure of the angle $A$.

 

[kaliprasad]

For the triangle with angles $A, B, C$, the following trigonometric equality holds.

$$\sin^2 B +\sin^2 C-\sin^2 A=\sin B \sin C$$

Find the measure of the angle $A$.

Spoiler

using law of sin's sin A = ka, sin B= kb, sin C = kc

we get
b^2+c^2 - a^2 = bc

or a^2 = b^2 + c^2 + bc (1)

by law of cos

a^2 = b^2 + c^2 - 2bc cos A (2)

from (1) and (2)

2 cos A = - 1 or cos A = - 1/2 or A = 120 degrees

 

[MarkFL]

Spoiler

using law of sin's sin A = ka, sin B= kb, sin C = kc

we get
b^2+c^2 - a^2 = bc

or a^2 = b^2 + c^2 + bc (1)

by law of cos

a^2 = b^2 + c^2 - 2bc cos A (2)

from (1) and (2)

2 cos A = - 1 or cos A = - 1/2 or A = 120 degrees

Your technique is nice and elegant, but you have made a simple sign error...:D

 

[anemone]

Spoiler

using law of sin's sin A = ka, sin B= kb, sin C = kc

we get
b^2+c^2 - a^2 = bc

or a^2 = b^2 + c^2 + bc (1)

by law of cos

a^2 = b^2 + c^2 - 2bc cos A (2)

from (1) and (2)

2 cos A = - 1 or cos A = - 1/2 or A = 120 degrees

Well done, kaliprasad...though you have made a minor mistake because $\ang A = 60 ^{\circ}$...:p

Also, I think my silly method is not worth mentioning after reading to your method!

 

[kaliprasad]

Well done, kaliprasad...though you have made a minor mistake because $\ang A = 60 ^{\circ}$...:p

Also, I think my silly method is not worth mentioning after reading to your method!

thanks markFL and anemone

it should be

Spoiler

a^2 = b^2 + c^2 - bc (1)

by law of cos

a^2 = b^2 + c^2 - 2bc cos A (2)

from (1) and (2)

2 cos A = 1 or cos A = 1/2 or A = 60 degrees

sorry for my mistake

 

[MarkFL]

My solution:

Spoiler

Since $$A=\pi-(B+C)$$ and $$\sin(\pi-\theta)=\sin(\theta)$$

The equation becomes:

$$\sin^2(B)+\sin^2(C)-\sin^2(B+C)=\sin(B)\sin(C)$$

Using the angle-sum identity for sine, we may write:

$$\sin^2(B)+\sin^2(C)-\sin^2(B)\cos^2(C)-2\sin(B)\sin(C)\cos(B)\cos(C)-\sin^2(C)\cos^2(B)=\sin(B)\sin(C)$$

$$\sin^2(B)\left(1-\cos^2(C) \right)+sin^2(C)\left(1-\cos^2(B) \right)=\sin(B)\sin(C)(1+2\cos(B)\cos(C))$$

Using Pythagorean identities, we have:

$$2\sin^2(B)\sin^2(C)=\sin(B)\sin(C)(1+2\cos(B)\cos(C))$$

Assuming the triangle is not degenerate, i.e., $$\sin(B)\sin(C)\ne0$$ we have:

$$2\sin(B)\sin(C)=1+2\cos(B)\cos(C)$$

$$-\frac{1}{2}=\cos(B)\cos(C)-\sin(B)\sin(C)=\cos(B+C)$$

$$B+C=120^{\circ}\implies A=60^{\circ}$$

 

[anemone]

thanks markFL and anemone

it should be

Spoiler

a^2 = b^2 + c^2 - bc (1)

by law of cos

a^2 = b^2 + c^2 - 2bc cos A (2)

from (1) and (2)

2 cos A = 1 or cos A = 1/2 or A = 60 degrees

sorry for my mistake

Don't be sorry, kali! And if you would like, you can make it up by joining me to post many interesting and challenging problems for the folks here...hehehe...(Emo)

My solution:

Spoiler

Since $$A=\pi-(B+C)$$ and $$\sin(\pi-\theta)=\sin(\theta)$$

The equation becomes:

$$\sin^2(B)+\sin^2(C)-\sin^2(B+C)=\sin(B)\sin(C)$$

Using the angle-sum identity for sine, we may write:

$$\sin^2(B)+\sin^2(C)-\sin^2(B)\cos^2(C)-2\sin(B)\sin(C)\cos(B)\cos(C)-\sin^2(C)\cos^2(B)=\sin(B)\sin(C)$$

$$\sin^2(B)\left(1-\cos^2(C) \right)+sin^2(C)\left(1-\cos^2(B) \right)=\sin(B)\sin(C)(1+2\cos(B)\cos(C))$$

Using Pythagorean identities, we have:

$$2\sin^2(B)\sin^2(C)=\sin(B)\sin(C)(1+2\cos(B)\cos(C))$$

Assuming the triangle is not degenerate, i.e., $$\sin(B)\sin(C)\ne0$$ we have:

$$2\sin(B)\sin(C)=1+2\cos(B)\cos(C)$$

$$-\frac{1}{2}=\cos(B)\cos(C)-\sin(B)\sin(C)=\cos(B+C)$$

$$B+C=120^{\circ}\implies A=60^{\circ}$$

Bravo, MarkFL! And I like your method as well! Just so you know, my method is so much longer and more tedious than yours! (Angry)

 

[MarkFL]

...And if you would like, you can make it up by joining me to post many interesting and challenging problems for the folks here...hehehe...(Emo)...

(Rofl) Now there's the ultimate challenge! (Wave) (Nerd)

 

[anemone]

(Rofl) Now there's the ultimate challenge! (Wave) (Nerd)

Very funny, MarkFL!(Rofl)