Question

2 cos 37° 30× cos 7° 30​

Answer 1

$2\cos37^\circ30'\cdot\cos7^\circ30'=2\cdot\dfrac{1}{2}\left(\cos(37^\circ30'-7^\circ30')+\cos(37^\circ30'+7^\circ30')\right)=\\=\cos30^\circ+\cos45^\circ=\dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{2}}{2}=\dfrac{\sqrt{3}+\sqrt{2}}{2}$

Answer 2

To calculate the value of the expression "2 cos 37° 30 × cos 7° 30," we'll need to use the trigonometric identity for the product of two cosines:

cos(A) × cos(B) = 1/2 [cos(A + B) + cos(A - B)]

Let's substitute the given values into this identity:

A = 37° 30 B = 7° 30

cos(A) × cos(B) = 1/2 [cos(A + B) + cos(A - B)] cos(37° 30) × cos(7° 30) = 1/2 [cos(37° 30 + 7° 30) + cos(37° 30 - 7° 30)]

Now we can calculate the values inside the brackets:

37° 30 + 7° 30 = 45° 37° 30 - 7° 30 = 30°

Let's calculate the cosine of these angles:

cos(45°) = √2/2 cos(30°) = √3/2

Substituting these values back into the equation:

cos(37° 30) × cos(7° 30) = 1/2 [cos(45°) + cos(30°)] cos(37° 30) × cos(7° 30) = 1/2 [√2/2 + √3/2]

Now let's simplify further:

cos(37° 30) × cos(7° 30) = 1/2 [(√2 + √3)/2]

Finally, multiplying through by 2:

cos(37° 30) × cos(7° 30) = (√2 + √3)/2

Therefore, the value of the expression "2 cos 37° 30 × cos 7° 30" is (√2 + √3)/2.