4 sin5° sin55 sin65º has the values equal to
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Answers
Answer:
The value of 4sin5° sin55° sin65° = sin 15° - 2sin5°
Step-by-step explanation:
The value of 4sin5° sin55° sin65°
4sin5° sin55° sin65°
By the trigonometric product formulas
sinx° siny° = [cos(x-y)-cos(x+y)]/2
sin55° sin65° = [cos(55-65)°-cos(55+65)°]/2
= [cos(-10)°-cos(120)°]/2
= [cos(10)°-cos(90+30)°]/2
= [cos10°-cos30°]/2
sin55° sin65° = [cos10°-(1//2)]/2
now substitute the above value in 4sin5° sin55° sin65°
4sin5° sin55° sin65° = 4sin5° {[cos10°-(1//2)]/2}
= 2sin5°cos10° - 2(1/2)sin5°
= 2sin5°cos10° - sin5°
we know that sinx cosy = [sin(x+y)+sin(x-y)]/2
then sin5°cos10° = [sin(5°+10°)+sin(5°-10°)]/2
= [sin15°+sin(-5)°]/2
= [sin15°-sin5°]/2
substitute the above value in 2sin5°cos10° - sin5°
4sin5° sin55° sin65° = 2sin5°cos10° - sin5°
= 2{[sin15°-sin5°]}/2 -sin5°
= sin15°-sin5°-sin5° °
= sin 15° - 2sin5°
Hence, the value of 4sin5° sin55° sin65° = sin 15° - 2sin5°.
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Answer:
The value of 4sin5° sin55° sin65° is sin115° + sin15° - sin65°.
Step-by-step explanation:
Recall the identities,
(i) -2sinA sinB = cos(A + B) - cos(A - B)
This implies, 2sinA sinB = cos(A - B) - cos (A + B).
(ii) 2cosA sinB = sin(A + B) - sin(A - B)
To find:-
The value of 4sin5° sin55° sin65°.
Consider the given trigonometric function as follows:
4sin5° sin55° sin65°
Rewrite as follows:
2(2sin5° sin55°) sin65°
Using the identity (i), simplify the function inside the bracket as follows:
⇒ 2(cos(5 - 55)° - cos(5 + 55)°) sin65°
⇒ 2(cos(-50°) - cos60°) sin65°
Since cos(-x) = cosx
⇒ 2(cos50° - cos60°) sin65°
⇒ 2cos50° sin65° - 2cos60° sin65°
⇒ 2cos50° sin65° - 2(1/2) sin65° (Since cos60° = 1/2)
⇒ 2cos50° sin65° - sin65°
Now, using the identity (ii) to simplify further.
⇒ [sin(50 + 65)° - sin(50 - 65)°] - sin65°
⇒ [sin115° - sin(-15)°] - sin65°
As we know,
sin(-x) = -sinx
⇒ [sin115° + sin15°] - sin65°
⇒ sin115° + sin15° - sin65°
Therefore, the value of 4sin5° sin55° sin65° is sin115° + sin15° - sin65°.
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