Sin(30) = 1/2 Sin(45) = root(2)/2 Sin(60) = root(3)/2 Cos(30) = root(3)/2 Cos(45) = root(2)/2 Cos(60) = 1/2 Tan(30) = root(3)/3 Tan(45) = 1 Tan(60) = root(3) Csc(30) = 2 Csc(45) = root(2) Csc(60) = 2root(3)/3 Sec(30) = 2root(3)/3 Sec(45) = root(2) Sec(60) = 2 Cot(30) = root(3) Cot(45) = 1 Cot(60) = root(3)/3
one over root of 2 or (1/square root of 2) or 1/1.414213562 or 0.707106781
sin x + csc x = 2 sin x + 1/sin x = 2 (sin2 x + 1)/sin x = 2/1 (cross multiply) sin2 x + 1 = 2sin x (subtract 2sin x to both sides) sin2 x - 2sinx + 1 = 0 (sin x - 1)2 = 0 (take the square root of both sides) sin x - 1 = 0 (add 1 to both sides) sin x = 1 x = sin-1 1 = 90⁰ 40x = 40(90⁰) = 3,600⁰ sin 40x = sin (3,600⁰ ) = 0 50x = 50(90⁰) = 4500⁰ sin 50x = sin (4,500⁰) = 0, so that csc 50x is undefined, we cannot divide by 0. Then, sin40x + csc50x is undefined.
cosine 45° = √2/2 (Square root of 2 over 2)
There are two ways to solve for the double angle formulas in trigonometry. The first is to use the angle addition formulas for sine and cosine. * sin(a + b) = sin(a)cos(b) + cos(a)sin(b) * cos(a + b) = cos(a)cos(b) - sin(a)sin(b) if a = b, then * sin(2a) = sin(a)cos(a) + cos(a)sin(a) = 2sin(a)cos(a) * cos(2a) = cos2(a) - sin2(b) The cooler way to solve for the double angle formulas is to use Euler's identity. eix = cos(x) + i*sin(x). Yes, that is "i" as in imaginary number. we we put 2x in for x, we get * e2ix = cos(2x) + i*sin(2x) This is the same as * (eix)2 = cos(2x) + i*sin(2x) We can substitute our original equation back in for eix. * (cos(x) + i*sin(x))2 = cos(2x) + i*sin(2x) We can distribute the squared term. * cos2(x) + i*sin(x)cos(x) + i*sin(x)cos(x) + (i*sin(x))2 = cos(2x) + i*sin(2x) And simplify. Because i is SQRT(-1), the i squared term becomes negative. * cos2(x) + 2i*sin(x)cos(x) - sin2(x) = cos(2x) + i*sin(2x) * cos2(x) - sin2(x) + 2i*sin(x)cos(x) = cos(2x) + i*sin(2x) Now you can plainly see both formulas in the equation arranged quite nicely. I don't yet know how to get rid of the i, but I'm working on it.
one over the square root of 2 or 0.850903525
1.5
sin-30 = (-1) x 1/(square root of 2) -sin30 = -(1/square root of 2) They are equal
Do sin(x), square it, and then multiply it by two.
36
The derivative of cos x is -sin x, the derivative of square root of x is 1/(2 root(x)). Applying the chain rule, the derivative of cos root(x) is -sin x times 1/(2 root(x)), or - sin x / (2 root x).
sin(405) = square root of 2 divided by 2 which is about 0.7071067812
square root 3/2
1/square root 2
sin(1,305) = sin(225) = -0.70711 (rounded) = 1/2 of the negative square root of 2.
PLUS 2
It is not totally clear to what the square root applies*; if just the 2, then: d/dx ((√2)sin x) = (√2) cos x if all of 2 sin x, then: d/dx (√(2 sin x)) = cos x / √(2 sin x) * for the second version I would expect "square root all of 2 sin x" but some people would write as given in the question meaning this, so I've given both just in case.