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Answer:
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Answer:
1.cos 75°
= We need to find the value of cos 75o
cos 75o = cos (30+45)
=The values of sin 45, sin 30, cos 45 and =cos 30 are commonly known.
=The values of cos and sin 30 also cos and sin 345 are listed below
=cos 30 = √3 /2
=sin 30 = 1/2,
=sin 45 = cos 45 = 1/√2
=We use these to determine the value of cos 75 and sin 75.
=We know the identity
=cos (x + y) = (cos x) (cos x) – (sin x)(sin y)
=cos 75 = cos (30 + 45)
=cos 75 = (cos 30)(cos 45) – (sin 30)(sin 45)
=cos 75 = √3/2 × 1/√2 – 1/2 × 1/√2
=cos75o=122√(3–√−1)
2. sin 15°
= Value of sine 15 degrees can be evaluated easily. ...
=The actual value of sin 15 degrees is given by: ...
=Sin P/2 + Cos P/2 = ± √ (1 + sin P) ...
=Sin 15° + Cos 15° = ±√ (1 + sin 30) …( ...
=Sin P/2 – Cos P/2 = ± √(1 – sin P) ...
=Sin 15° – Cos 15° = ±√(1 – sin 30°) …( ...
=Since, we know, Sin 15° = (√3–1)/2√2.
3. sin 75°
=(√3 + 1)/ 2√2
Sin 75 we can write it as
Sin 75 = Sin(45+30)…………………..(1)
By applying the formula
Sin (A + B) = Sin A. Cos B + Cos A. Sin B
Sin (45 + 30) = Sin 45. Cos 30 + Cos 45. Sin 30…………………..(2)
Sin Values
sin 0° = √(0/4) = 0
sin 30° = √(1/4) = ½
sin 45° = √(2/4) = 1/√2
sin 60° = √3/4 = √3/2
cos 90° = √(4/4) = 1
Cos Values
cos 0° = √(4/4) = 1
cos 30° = √(3/4) = √3/2
cos 45° = √(2/4) = 1/√2
cos 60° = √(1/4) = 1/2
cos 90° = √(0/4) = 0
Substitute the value of sin 30, sin 45, cos 30 and cos 45 degrees, then equation (2) will becomes
Sin (45 + 30) = Sin 45. Cos 30 + Cos 45. Sin 30
Sin (45 + 30) = 1/√2 . √3/2 + 1/√2 . 1/2
Sin (45 + 30) = (√3 + 1) / 2√2
Hence the value of Sin 75 degree is equal to (√3 + 1) / 2√2.
4. cos 105°
= cos (60° + 45°)
cos (A + B) = cos A cos B - sin A sin B
cos (60 + 45) = cos 60° cos 45° - sin 60° sin 45° ----(1)
sin 45° = 1/√2
sin 60 = √3/2
cos 45° = 1/√2
cos 60° = 1/2
By applying the above values in the first equation, we get
cos (60 + 45) = (1/2) (1/√2) - (√3/2)(1/√2)
= (1/2√2) - (√3/2√2)
= (1 - √3)/2√2
So, the value of cos 105° is (1 - √3)/2√2.