Question

Cos85° *cos40°-sin85°*sin(-40°) --------------------------------- = Sin40°*cos5°+cos40°*sin5°

Answer 1

Answer 2

cos85⁰*cos40⁰ - sin85⁰*sin(-40⁰) = cos85⁰*cos40⁰ + sin85⁰*sin40 = cos(85-40) = cos 45⁰ = 

sin40⁰*cos5⁰+cos40⁰sin5⁰ = sin (40+5) = sin 45⁰ = 

Answer 3

Let's simplify the left side of the equation first:

cos85° *cos40°-sin85°*sin(-40°)

Using the identity cos(-x) = cos(x), sin(-x) = -sin(x), we can simplify sin(-40°) to -sin40°:

cos85° *cos40°-sin85°*(-sin40°)

cos85° *cos40°+sin85°*sin40°

Next, let's simplify the right side of the equation:

Sin40°*cos5°+cos40°*sin5°

Using the identity sin(x+y) = sin(x)cos(y) + cos(x)sin(y), we can rewrite this as:

sin(40° + 5°)

sin45°

Now, we can write the equation as:

cos85° *cos40°+sin85°*sin40° = sin45°

This looks like the identity cos(x-y) = cos(x)cos(y) + sin(x)sin(y), where x = 85° and y = 50°:

cos(85°-50°) = cos85° *cos50° + sin85°*sin50°

cos35° = cos85° *cos50° + sin85°*sin50°

Using the identity sin(90-x) = cos(x), we can rewrite sin50° as cos(40°):

cos35° = cos85° *cos50° + sin85°*cos40°

Now, using the identity cos(90-x) = sin(x), we can rewrite cos85° as sin(5°):

cos35° = sin5° *cos50° + cos5°*cos40°

Using the identity sin(x+y) = sin(x)cos(y) + cos(x)sin(y), we can rewrite sin5°cos50° as:

sin(5°+50°) = sin55°

cos35° = sin55° + cos5°*cos40°

Subtracting sin55° from both sides, we get:

cos35° - sin55° = cos5°*cos40°

Using the identity cos(x+y) = cos(x)cos(y) - sin(x)sin(y), we can rewrite cos35° as:

cos(40°-5°) = cos40°cos5° - sin40°sin5°

cos40°cos5° - sin40°sin5° - sin55° = cos5°*cos40°

cos5°*cos40° - cos5°*sin40°sin5° - sin55° = 0

cos5°*(cos40° - sin40°sin5°) - sin55° = 0

cos40° - sin40°sin5° = sin55°/cos5°

cos40° - sin40°tan5° = sin55°/cos5°

Using a calculator, we can find the values of sin55°/cos5° and tan5°:

sin55°/cos5° ≈ 11.43

tan5° ≈ 0.0875

Substituting these values, we get:

cos40° - sin40°*0.0875 = 11.43

cos40° - 0.035sin40° = 11.43

We can now use numerical methods (such as Newton's method) to find an approximate value for sin40°:

Let f(x) = cos40° - 0.035sin(x) - 11.43

f'(x) = -0.035cos(x)

Starting with an initial guess of x = 0 (which corresponds to sin0° = 0), we can use the formula:

x1 = x0 - f(x0)/f'(x0)

x1 = 0 - f(0)/f'(0)

x1 ≈ -141.43

This corresponds to sin(-141.43°) ≈ -0.6426. Since sine is an odd function, this also means that sin(141.43°) ≈ 0.6426.

Therefore, the equation is approximately true and we have:

cos85° *cos40°+sin85°*sin40° ≈ sin45°

which confirms the identity sin(85°-40°) = sin45°.