Final answer:
To find the exact value of sin 255º x sin 75º, we use the identity sin(180º + x) = -sin(x) to transform sin 255º into -sin 75º, and then use the sine of a sum formula and standard angle values to calculate -sin²(75º), which equals -(4 + √12) / 8.
Step-by-step explanation:
The student's question is about finding the exact value of the expression sin 255º multiplied by sin 75º. To solve this, we can use trigonometric identities and properties of sine. The sine of an angle in the third quadrant, like 255º, is negative, and the sine of an angle in the first quadrant, like 75º, is positive.
The identity sin(180º + x) = -sin(x) can be applied to sin 255º since 255º is 180º plus an additional 75º. Thus, sin 255º = -sin(75º). The expression now becomes -sin(75º) × sin(75º), which simplifies to -sin²(75º).
Using a calculator or knowing that sin(75º) = sin(30º + 45º), we can apply the sum of angles identity for sine, which states sin(a + b) = sin(a)cos(b) + cos(a)sin(b), therefore, sin(75º) = sin(30º)cos(45º) + cos(30º)sin(45º). As both 30º and 45º are well-known angles, we can replace the sine and cosine values: sin(75º) = (1/2)√2/2 + (√3/2)(√2/2) = (√2 + √6)/4.
Sin²(75º) thus equals ((√2 + √6)/4)² = (√2 + √6)^2/16 = (2 + 2√12 + 6)/16 = 8 + 2√12/16 = 1/2 + √12/8. Multiplying this by -1 gives us the final exact value of the expression as -1/2 - √12/8 or -(4 + √12) / 8.