Question

Sin112sin67+cos67cos248=

Answer 1

$\displaystyle sin112sin67+cos67cos248=sin112sin67+cos67cos(360-112)=\\=sin112sin67+cos67cos112=cos(67-112)=-cos(-45)=cos45=\frac{\sqrt2}{2}$

Answer 2

To solve this trigonometric expression, we'll use the angle sum and difference identities for sine and cosine.

Angle sum identities:

1. sin(A + B) = sinAcosB + cosAsinB
2. cos(A + B) = cosAcosB - sinAsinB

Here, we have: sin(112)sin(67) + cos(67)cos(248)

Step 1: Convert degrees to radians (assuming the trigonometric functions take radians as arguments). 112 degrees = 112 * π / 180 radians ≈ 1.95 radians 67 degrees = 67 * π / 180 radians ≈ 1.17 radians 248 degrees = 248 * π / 180 radians ≈ 4.33 radians

Step 2: Apply the identities. sin(112)sin(67) + cos(67)cos(248) = sin(1.95) * sin(1.17) + cos(1.17) * cos(4.33)

Step 3: Calculate the values for sin(1.95), sin(1.17), cos(1.17), and cos(4.33). Note that these values can be computed using a calculator or lookup tables. sin(1.95) ≈ 0.932 sin(1.17) ≈ 0.923 cos(1.17) ≈ 0.385 cos(4.33) ≈ -0.436

Step 4: Substitute the values back into the expression: 0.932 * 0.923 + 0.385 * (-0.436)

Step 5: Calculate the result: 0.861 + (-0.168) ≈ 0.693

So, sin(112)sin(67) + cos(67)cos(248) ≈ 0.693.