Use an Addition or Subtraction Formula to find the exact value of the expression.
Sin(−5π/12)
some students have less difficulty if the angle is in degrees.
Let's try this approach ....
5π/12 radians = 75°
so sin(-5π/12)
= sin(-75°) which is in quadrant IV, making the sine negative
= -sin(75°)
= -sin(30° + 45°)
= -(sin30cos45 + cos30sin45)
= -( (1/2)(√2/2) + (√3/3)(√2/2) )
= - √2/4 - √6/4
so sin(05π/12) = -√2/4 - √6/4
check:
on my calculator:
sin(-5π/12) = -.9659...
and -√2/4 - √6/4 = -.9659..
My answer is correct
To find the exact value of the expression sin(-5π/12), we can use the addition or subtraction formula for sine.
The addition formula for sine states that sin(a - b) = sin(a) * cos(b) - cos(a) * sin(b).
In this case, we have sin(-5π/12), so we can rewrite it as sin(-π/3 - 2π/12).
Now, let's assign a = -π/3 and b = 2π/12.
Using these values in the formula, we have:
sin(-5π/12) = sin(-π/3 - 2π/12)
= sin(-π/3) * cos(2π/12) - cos(-π/3) * sin(2π/12)
To find the exact value, we need to know the values of sin(-π/3), cos(2π/12), cos(-π/3), and sin(2π/12).
Using the values from the unit circle, we have:
sin(-π/3) = -√3/2
cos(2π/12) = √3/2
cos(-π/3) = 1/2
sin(2π/12) = 1/2
Plugging these values back into the formula, we get:
sin(-5π/12) = (-√3/2)(√3/2) - (1/2)(1/2)
= -3/4 - 1/4
= -4/4
= -1
Therefore, the exact value of sin(-5π/12) is -1.
To find the exact value of sin(-5π/12), we can use the addition formula for sine.
The addition formula for sine states that sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
We can rewrite sin(-5π/12) using this formula as follows:
sin(-5π/12) = sin(π/3 - 2π/4)
Now let's apply the addition formula:
sin(π/3 - 2π/4) = sin(π/3)cos(2π/4) - cos(π/3)sin(2π/4)
Next, we just need to determine the exact values of sin(π/3), cos(2π/4), cos(π/3), and sin(2π/4). We can use the unit circle or commonly known values to find these.
Using the unit circle, we can determine the following values:
- sin(π/3) = √3/2
- cos(2π/4) = cos(π/2) = 0
- cos(π/3) = 1/2
- sin(2π/4) = sin(π/2) = 1
Now let's substitute these values back into the equation:
sin(π/3)cos(2π/4) - cos(π/3)sin(2π/4) = (√3/2)(0) - (1/2)(1)
Simplifying further:
(0) - (1/2) = -1/2
Therefore, the exact value of sin(-5π/12) is -1/2.