What is the exact value of sin(-15 degrees)?
Start with the anti-symmetry of sine
sin(x)=-sin(-x)
so sin(-15°)=-sin(15°)
Now use
cos²(x)-sin²(x)=cos(2x)
cos²(x)+sin²(x)=1
Subtract first equation from second:
2sin²(x)=(1-cos(2x))
sin(x)=sqrt((1-cos(2x))/2)
Put x=-15°
sin(15°)
=-sin(15°)
=-sqrt((1-cos(30°))/2)
=-sqrt((1-sqrt(3)/2)/2)
To find the exact value of sin(-15 degrees), we can use the angle addition and subtraction identity:
sin(-15) = sin(-30 + 15)
We know that sin(-30) = -1/2, which is a common trigonometric value. Therefore, we can use the angle subtraction identity:
sin(-30 + 15) = sin(-30)cos(15) + cos(-30)sin(15)
Since cos(-30) = cos(30) (cosine function is an even function), and sin(-30) = -sin(30) (sine function is an odd function), we can simplify the expression:
sin(-30 + 15) = -1/2 * cos(15) + √3/2 * sin(15)
Using the exact values of cos(15) and sin(15), which can be found using special triangles or trigonometric identities, we get:
sin(-30 + 15) = -1/2 * (√6 - √2)/4 + √3/2 * (√6 + √2)/4
Simplifying further, we have:
sin(-15) = -√6/8 + √2/8 + √18/8 + √6/8
Combining similar terms, we get:
sin(-15) = (√2 + √6 + √18)/8
So, the exact value of sin(-15 degrees) is (√2 + √6 + √18)/8.