sin^2(30+A)-sin^2(30-A)=√3/2sin20
Answers
The measure of A = 10°
Given :-
sin²(30+A) - sin²(30-A) = (√3/2) sin 20°
To find :-
The measure of A
Solution :-
Given that
sin²(30+A)° - sin²(30-A)° = (√3/2) sin 20°
We know that
(a+b)(a-b) = a²-b²
We have, In LHS ,
a = sin (30+A)°
b = sin (30-A)°
Now,
sin (30°+A+30°-A) sin (30°+A-30°+A)=(√3/2) sin 20°
=> sin (60°) sin (2A) = (√3/2) sin 20°
=> (√3/2) sin 2A = (√3/2) sin 20°
On comparing both sides then
2A = 20°
=> A = 20°/2
=> A = 10°
Therefore, The measure of A = 10°
Alternative Method:-
Given that
sin²(30+A)° - sin²(30-A)° = (√3/2) sin 20°
We know that
sin (C+D) = sin C cos D + cos C sin D
now
sin (30+A)
= sin 30° cos A + cos 30° sin A
= (1/2) cos A + (√3/2) sin A
= (cos A + √3 sin A)/2
Now,
sin² (30+A)
= [(cos A + √3 sin A)/2]²
= (cos² A + 3 sin² A + 2√3 sin A cos A]/4
= ( cos² A + 3 sin² A + √3 sin 2A)/4
and
We know that
sin (C-D) = sin C cos D - cos C sin D
now
sin (30-A)
= sin 30° cos A - cos 30° sin A
= (1/2) cos A - (√3/2) sin A
= (cos A -√3 sin A)/2
Now,
sin² (30-A)
= [(cos A - √3 sin A)/2]²
= (cos² A + 3 sin² A - 2√3 sin A cos A]/4
= ( cos² A + 3 sin² A - √3 sin 2A)/4
Now,
sin²(30+A)° - sin²(30-A)°
= [( cos² A + 3 sin² A +√3 sin 2A)/4] - [ ( cos² A + 3 sin² A - √3 sin 2A)/4 ]
= (√3 sin 2A + √3 sin 2A)/4
= 2√3 sin 2A/4
= (√3 sin 2 A)/2
= (√3/2) sin 2A
Now
Given equation becomes
(√3/2) sin 2A = (√3/2) sin 20°
On comparing both sides then
2A = 20°
=> A = 20°/2
=> A = 10°
Therefore, The measure of A = 10°
Answer :-
The measure of A is 10°
Used formulae:-
• (a+b)(a-b) = a²-b²
•(a+b)² = a²+2ab+b²
• (a-b)² = a²-2ab+b²
• (a+b)²-(a-b)² = 4ab
• sin 60° = √3/2
• sin (C+D) = sin C cos D + cos C sin D
• sin (C-D) = sin C cos D - cos C sin D
• sin 2A = 2 sin A cos A
Answer:
Step-by-step explanation:
sin^2 30 means sin 30*sin 30 = (1/3^0.5)*(1/3^0.5) = 1/3
sin 30^2 means sin 900. Divide 900/180 = 5. Or sin 30^2= sin 900 = sin 5(pi) = 0.