Question

Cos(36 +a)cos(54+a)-sin(36+a)sin(54+a)

Answer 1

Упрошаем( cos(t)cos(s)-sin(t)sin(s)=cos(t+s))и получаем cos(36+a+54+a)=cos(90+a+a)=cos(90+2a)

Answer 2

To simplify the expression, we'll use the trigonometric identities for the sum of angles:

1. Cosine of the sum of two angles: cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

2. Sine of the sum of two angles: sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

Let's apply these identities to the given expression:

cos(36 + a)cos(54 + a) - sin(36 + a)sin(54 + a)

Using the cosine of the sum of angles identity, we get:

= [cos(36)cos(a) - sin(36)sin(a)][cos(54)cos(a) - sin(54)sin(a)]

Now, using the sine of the sum of angles identity, we get:

= [(cos(36)cos(a))(cos(54)cos(a)) - (sin(36)sin(a))(cos(54)cos(a))]

• [(cos(36)cos(a))(sin(54)sin(a)) + (sin(36)sin(a))(sin(54)sin(a))]

Now, let's simplify each term:

= [cos(36)cos(54)cos^2(a) - sin(36)sin(54)cos^2(a)]

• [cos(36)cos(54)sin^2(a) + sin(36)sin(54)sin^2(a)]

Since cos^2(a) + sin^2(a) = 1, we can rewrite the expression:

= cos(36)cos(54) - sin(36)sin(54)

Now, you might recognize this as the cosine of the difference of two angles identity:

cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

Thus:

cos(36 - 54) = cos(-18)

Finally, the value of cos(-18) can be calculated using standard trigonometric values:

cos(-18) = cos(18) = 0.9510565163 (approx)

So the simplified expression is approximately 0.9510565163.