Question

Cos78+cos18 тригонометрия решение

Answer 1

Cosx+cosy=2.cos(x+y)/2.cos (x-y)/2
cos78+cos18=2.cos(78+18)/2.cos (78-18)/2=2.cos48.cos30=2.V3/2.cos48=V3.cos48

Answer 2

Cos78+cos18=2 сos(78+18)cos(78-18)=2  Cos48cos30= 2  сos48√3/2 = √3 сos48 °

Answer 3

To solve the expression cos(78°) + cos(18°), we can use the trigonometric identity:

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

Using this identity, we can rewrite the expression as follows:

cos(78°) + cos(18°) = cos(60° + 18°) + cos(18°)

Now, let's use the trigonometric identity mentioned above to simplify further:

cos(78°) + cos(18°) = [cos(60°)cos(18°) - sin(60°)sin(18°)] + cos(18°)

Since cos(60°) = 0.5 and sin(60°) = √3/2, we can substitute these values in:

cos(78°) + cos(18°) = [0.5 * cos(18°) - (√3/2) * sin(18°)] + cos(18°)

Combining like terms, we have:

cos(78°) + cos(18°) = 1.5 * cos(18°) - (√3/2) * sin(18°)

At this point, we need to know the exact values of cos(18°) and sin(18°). In trigonometry, we can determine these values using various methods such as geometric constructions or numerical approximations.

For this calculation, we'll use the fact that cos(18°) = (√5 - 1) / 4 and sin(18°) = (√10 - √2) / 4, which can be derived from the golden ratio.

Substituting these values into the equation:

cos(78°) + cos(18°) = 1.5 * [ (√5 - 1) / 4 ] - (√3/2) * [ (√10 - √2) / 4 ]

Simplifying further:

cos(78°) + cos(18°) = (3√5 - 3) / 8 - (√30 - √6) / 8

Finally, combining the terms with the same denominator:

cos(78°) + cos(18°) = (3√5 - √30 - 3 + √6) / 8

Therefore, the simplified value of cos(78°) + cos(18°) is (3√5 - √30 - 3 + √6) / 8.