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It helps to convert this kind of equation into one that has only sines and cosines, by using the basic definitions of the other functions in terms of sines and cosines.
sin x / (1 - cos x) = csc x + cot x
sin x / (1 - cos x) = 1 / sin x + cos x / sin x
Now it should be easy to do some simplifications:
sin x / (1 - cos x) = (1 + cos x) / sin x
Multiply both sides by 1 + cos x:
sin x (1 + cos) / ((1 - cos x)(1 + cos x)) = (1 + cos x)2 / sin x
sin x (1 + cos) / (1 - cos2x) = (1 + cos x)2 / sin x
sin x (1 + cos) / sin2x = (1 + cos x)2 / sin x
sin x (1 + cos x) / sin x = (1 + cos x)2
1 + cos x = (1 + cos x)2
1 = 1 + cos x
cos x = 0
So, cos x can be pi/2, 3 pi / 2, etc.
In some of the simplifications, I divided by a factor that might be equal to zero; this has to be considered separately. For example, what if sin x = 0? Check whether this is a solution to the original equation.
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If a cos theta plus b sin theta equals 8 and a sin theta - b cos theta equals 5 show that a squared plus b squared equals 89?
Well, darling, if we square the first equation and the second equation, add them together, and do some algebraic magic, we can indeed show that a squared plus b squared equals 89. It's like a little math puzzle, but trust me, the answer is as sassy as I am.
What is the length of side c in Triangle ABC to the nearest whole number if A equals 42 degrees and B equals 87 degrees and a equals 24?
28 The Law of Sines: a/sin A = b/sin B = c/sin C 24/sin 42˚ = c/sin (180˚ - 42˚ - 87˚) since there are 180˚ in a triangle. 24/sin 42˚ = c/sin 51˚ c = 24(sin 51˚)/sin 42˚ ≈ 28
1 over sin x equals what?
1/sin x = csc x
How do you prove that 2 sin 3x divided by sin x plus 2 cos 3x divided by cos x equals 8 cos 2x?
You need to know the trigonometric formulae for sin and cos of compound angles. sin(x+y) = sin(x)*cos(y)+cos(x)*sin(y) and cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y) Using these, y = x implies that sin(2x) = sin(x+x) = 2*sin(x)cos(x) and cos(2x) = cos(x+x) = cos^2(x) - sin^2(x) Next, the triple angle formulae are: sin(3x) = sin(2x + x) = 3*sin(x) - 4*sin^3(x) and cos(3x) = 4*cos^3(x) - 3*cos(x) Then the left hand side = 2*[3*sin(x) - 4*sin^3(x)]/sin(x) + 2*[4*cos^3(x) - 3*cos(x)]/cos(x) = 6 - 8*sin^2(x) + 8cos^2(x) - 6 = 8*[cos^2(x) - sin^2(x)] = 8*cos(2x) = right hand side.
Tan equals 0.3421 sin equals 0.3237 Which quadrant does it terminate?
The value of tan and sin is positive so you must search quadrant that tan and sin value is positive. The only quadrant fill that qualification is Quadrant 1.
