find the exact value of sin 11pi/12 using the sum or difference formula.
Some students feel more comfortable working with degrees.
sin(11π/12) = sin 165° = sin 15° by the CAST rule
sin15° = sin(45°-30°)
= sin45°cos30° - cos45°sin30°
= (√2/2)(√3/2) - (√2/2)(1/2)
= (√6 - √2)/4
sin 11π/12 = (√6-√2)/4
11/12 = 2/12 + 9/12 = 1/6 + 3/4
sin pi/6 = 1/2
cos pi/6 = (1/2) sqrt 3
sin 3 pi/4 = (1/2)sqrt 2
cos 3 pi/4 = -(1/2) sqrt 2
To find the exact value of sin(11π/12) using the sum or difference formula, let's first express 11π/12 as the sum or difference of two angles for which the values of sine are known.
We know that π/12 = π/4 - π/6. Thus, we can rewrite 11π/12 as:
11π/12 = π/4 + (3π/4 - π/6)
Next, we can use the sum or difference formula for sine: sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
Applying this formula to our expression, we have:
sin(11π/12) = sin(π/4 + (3π/4 - π/6))
= sin(π/4)cos(3π/4 - π/6) + cos(π/4)sin(3π/4 - π/6)
Since we know the values of sine and cosine of π/4, π/6, and 3π/4, we can substitute these values into the expression:
sin(11π/12) = (sqrt(2)/2)cos(3π/4 - π/6) + (sqrt(2)/2)sin(3π/4 - π/6)
Now, let's simplify each term:
cos(3π/4 - π/6)
We can use the cosine identity:
cos(α - β) = cos(α)cos(β) + sin(α)sin(β)
Applying this formula, we have:
cos(3π/4 - π/6) = cos(3π/4)cos(π/6) + sin(3π/4)sin(π/6)
= (-sqrt(2)/2)(√3/2) + (-sqrt(2)/2)(1/2)
= -√6/4 - √2/4
= -(√6 + √2)/4
Similarly,
sin(3π/4 - π/6) = sin(3π/4)cos(π/6) - cos(3π/4)sin(π/6)
= (sqrt(2)/2)(√3/2) - (-sqrt(2)/2)(1/2)
= √6/4 - √2/4
= (√6 - √2)/4
Now, substituting these values back into the equation:
sin(11π/12) = (sqrt(2)/2)(-(√6 + √2)/4) + (sqrt(2)/2)((√6 - √2)/4)
= -√6/8 - √2/8 + √6/8 - √2/8
= (-√6 - √2 + √6 - √2)/8
= -2√2/8
= -√2/4
Therefore, the exact value of sin(11π/12) using the sum or difference formula is -√2/4.
To find the exact value of sin(11π/12) using the sum or difference formula, we can break it down into two angles for which we know the exact values of sine functions. We can express 11π/12 as the sum of two angles by using the following property: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
Let's break down 11π/12 as the sum of two angles. We know that π/6 + π/4 = 5π/12. Therefore, we can rewrite 11π/12 as 5π/12 + 6π/12.
Now, using the sum formula, sin(11π/12) can be expressed as sin(5π/12 + 6π/12).
Applying the sum formula, we get:
sin(11π/12) = sin(5π/12)cos(6π/12) + cos(5π/12)sin(6π/12).
Since we know the exact values of sin(π/4) = √2/2 and sin(π/6) = 1/2, we can rewrite the equation as:
sin(11π/12) = (1/2) cos(π/2) + (√2/2) sin(π/2).
Since cos(π/2) = 0 and sin(π/2) = 1, we have:
sin(11π/12) = (1/2)(0) + (√2/2)(1).
Finally, simplifying the equation gives us the exact value of sin(11π/12):
sin(11π/12) = √2/2.