Question

Cos 157° cos 97°+ sin 157° sin 97°

Answer 1

Cos 157° cos 97°+ sin 157° sin 97° - формула cos разности, т.е. cos(157-97)=cos(60)=1/2

Answer 2

To evaluate the expression cos 157° cos 97° + sin 157° sin 97°, we can use the trigonometric identity for the cosine of the sum of two angles:

cos(A + B) = cos A * cos B - sin A * sin B

In this case, A = 157° and B = 97°. So, the expression becomes:

cos(157° + 97°) = cos 157° * cos 97° - sin 157° * sin 97°

Now, we need to find the values of cos 157° and cos 97° and sin 157° and sin 97°. To do that, we can use the following trigonometric identities:

cos(180° - θ) = -cos θ sin(180° - θ) = sin θ

cos(157°) = cos(180° - 157°) = -cos 23° cos(97°) = cos(180° - 97°) = -cos 83°

sin(157°) = sin(180° - 157°) = sin 23° sin(97°) = sin(180° - 97°) = sin 83°

Now, we can substitute these values back into the original expression:

cos 157° cos 97° + sin 157° sin 97° = (-cos 23°) * (-cos 83°) + (sin 23°) * (sin 83°)

Since cos(θ) * cos(θ) = cos^2(θ) and sin(θ) * sin(θ) = sin^2(θ), we have:

= cos^2(23°) + sin^2(23°)

The trigonometric identity cos^2(θ) + sin^2(θ) = 1 states that for any angle θ, the sum of the squares of the cosine and sine of that angle is equal to 1. Therefore, the final result is:

= 1