Question

(cos 23pi/12-sin 11pi/12)*(sin 17pi/12-cos19pi/12)

Answer 1

(cos23π/12-sin11π/12)(sin17π/12-cos19π/12)
=(cos(2π-π/12)-sin(π-π/12))*
(sin(π+5π/12)-cos(2π-5π/12)=
(cosπ/12-sinπ/12)(-sin5π/12-cos5π/12)=
(cosπ/12-sinπ/12)(-sin(π/2-π/12)-
cos(π/2-π/12))==
(cosπ/12-sinπ/12)(-1)(cosπ/12+sinπ/12)=
-(cos²π/12-sin²π/12)=-cosπ/6=-√3/2

Answer 2

To find the value of the expression (cos(23π/12) - sin(11π/12)) * (sin(17π/12) - cos(19π/12)), we'll break it down into simpler trigonometric expressions and then calculate each part step by step.

Step 1: Convert angles to their reference angles in the first quadrant. We can use the periodicity of trigonometric functions to convert the given angles to their reference angles in the first quadrant:

• 23π/12 is in the fourth quadrant, and its reference angle in the first quadrant is 23π/12 - 2π = 11π/12.
• 11π/12 is also in the fourth quadrant, and its reference angle in the first quadrant is 11π/12 - π = π/12.
• 17π/12 is in the third quadrant, and its reference angle in the first quadrant is 17π/12 - π = 5π/12.
• 19π/12 is in the third quadrant, and its reference angle in the first quadrant is 19π/12 - π = 7π/12.

Now, the expression becomes:

(cos(11π/12) - sin(π/12)) * (sin(5π/12) - cos(7π/12)).

Step 2: Calculate the trigonometric values of the reference angles. We can now evaluate the trigonometric values of the reference angles:

• cos(11π/12) and cos(7π/12) are both values in the first quadrant, so they are positive.
• sin(π/12) and sin(5π/12) are both values in the first quadrant, so they are positive.

Step 3: Calculate the individual products. Now, let's calculate the individual products:

cos(11π/12) - sin(π/12) = cos(11π/12) - sin(π/12)

sin(5π/12) - cos(7π/12) = sin(5π/12) - cos(7π/12)

Step 4: Calculate the final result. Finally, we multiply the results of the individual products:

(cos(11π/12) - sin(π/12)) * (sin(5π/12) - cos(7π/12)) = (cos(11π/12) - sin(π/12)) * (sin(5π/12) - cos(7π/12))

Since we can't simplify it further without more precise numerical values, this is the final result for the given expression.