{sin(π-a) + sin(π/2 -a)}²
= {(sin(π)cos(a) - cos(π)sin(a)) + (sin(π/2)cos(a) - cos(π/2)sin(a))}²
= {(0*cos(a) - (-1)sin(a)) + (1*cos(a) - 0*sin(a))}²
= (sin(a) + cos(a))²
The third line, the sin(pi) became 0 and the cos(pi) became (-1) why?
I can understand the formula to solve this is: sin(x)cos(y) -cos(x)sin(y), but I can't understand what exactly happens there.
Much appreciated if anybody could clarify this for me!
= {(sin(π)cos(a) - cos(π)sin(a)) + (sin(π/2)cos(a) - cos(π/2)sin(a))}²
= {(0*cos(a) - (-1)sin(a)) + (1*cos(a) - 0*sin(a))}²
= (sin(a) + cos(a))²
The third line, the sin(pi) became 0 and the cos(pi) became (-1) why?
I can understand the formula to solve this is: sin(x)cos(y) -cos(x)sin(y), but I can't understand what exactly happens there.
Much appreciated if anybody could clarify this for me!
-
sin(pi) = sin(pi - 0) = sin0 = 0 [lying in the second quadrant where all values of sine are positive]
Similarly,
cos(pi) = cos(pi - 0) = - cos0 = - 1
[lying in the second quadrant where all values of cosine are negative].
Similarly,
cos(pi) = cos(pi - 0) = - cos0 = - 1
[lying in the second quadrant where all values of cosine are negative].
-
Draw a unit circle, and draw the angle pi radians. Since that's half a circle, it should be in the -x direction, and should slice the unit circle at the point (-1,0).
Every point on the unit circle is, by definition of sine and cosine, at the point (cos t, sin t), where t is the angle from the +x-axis....to the origin....and to the point on the unit circle you're talking about. In this case, those points are (1,0), (0,0), and (-1, 0), forming a 180 degree/pi radian line.
So you have cos pi = -1 and sin pi = 0.
There's a couple of more simplifications you might want to make....I don't know what kind of answer "they" wanted you to come up with. But you can foil (sin a + cos a)^2. After that, you can do two simplifications: use the pythagorean identity, and...if you know it....use a double angle identity. (Remember, identities work in BOTH directions!)
Every point on the unit circle is, by definition of sine and cosine, at the point (cos t, sin t), where t is the angle from the +x-axis....to the origin....and to the point on the unit circle you're talking about. In this case, those points are (1,0), (0,0), and (-1, 0), forming a 180 degree/pi radian line.
So you have cos pi = -1 and sin pi = 0.
There's a couple of more simplifications you might want to make....I don't know what kind of answer "they" wanted you to come up with. But you can foil (sin a + cos a)^2. After that, you can do two simplifications: use the pythagorean identity, and...if you know it....use a double angle identity. (Remember, identities work in BOTH directions!)
-
Think about what you know about the unit circle; every (x, y) on the circumference of the unit circle represents (cos(theta), sin(theta)), where theta is the angle whose terminal side passes through the point (x, y).
When theta = pi, (180 degrees), the terminal side is the negative x-axis and crosses the unit circle at the point (-1, 0). So the cosine of pi is -1, and the sine of pi is zero.
When theta = pi, (180 degrees), the terminal side is the negative x-axis and crosses the unit circle at the point (-1, 0). So the cosine of pi is -1, and the sine of pi is zero.