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pavadrin
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“The “t” Method”
Hey
Recently I have been studying for an upcoming test where it requires me to use “The “t” Method”. In this method the value of x for trigonometric equations is determined through vair the key component of “The “t” Method” is:
[tex]t=tan \frac{A}{2}[/tex]
[tex]tan A=\frac{1+t^2}{1-t^2}[/tex]
[tex]sin A=\frac{2t}{1+t^2}[/tex]
[tex]cos A=\frac{1-t^2}{1+t^2}[/tex]
If this is famliar to a reader by another name, could you please post this methods other name.
1. [tex]2sin x + 3cos x = 5[/tex]
2. [tex]3tan x + \sqrt{3}sec x=1[/tex]
3. [tex]10cos (pi x) + 3sin (2pi x)=4[/tex]
4. [tex]3sin 2x + 5cot 3x = 7[/tex]
5. [tex]csc x + 2sec (pi x)[/tex]
[tex]2\frac{2t}{1+t^2} + 3\frac{1-t^2}{1+t^2} = 5[/tex]
[tex]\frac{4t + 3 - 3t^2}{1+t^2} = 5[/tex]
[tex]4t + 3 -3t^2 = 5 +5t^2[/tex]
[tex]4t + 2t^2 = 2[/tex]
[tex]2t^2 + 4t - 2 = 0[/tex]
[tex]t = 0.4142135624[/tex] or [tex]t = -2.414213562[/tex]
[tex]t = \tan\frac{x}{2}[/tex]
[tex]x = 2tan^-1 0.4142135624[/tex] or [tex]x = 2tan^-1 -2.414213562[/tex]
[tex]x = 45[/tex] or [tex]x = -135.0005034[/tex]
Thanks well in advance,
Pavadrin
Hey
Recently I have been studying for an upcoming test where it requires me to use “The “t” Method”. In this method the value of x for trigonometric equations is determined through vair the key component of “The “t” Method” is:
[tex]t=tan \frac{A}{2}[/tex]
[tex]tan A=\frac{1+t^2}{1-t^2}[/tex]
[tex]sin A=\frac{2t}{1+t^2}[/tex]
[tex]cos A=\frac{1-t^2}{1+t^2}[/tex]
If this is famliar to a reader by another name, could you please post this methods other name.
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Problems which i need to solve:1. [tex]2sin x + 3cos x = 5[/tex]
2. [tex]3tan x + \sqrt{3}sec x=1[/tex]
3. [tex]10cos (pi x) + 3sin (2pi x)=4[/tex]
4. [tex]3sin 2x + 5cot 3x = 7[/tex]
5. [tex]csc x + 2sec (pi x)[/tex]
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My working for question 1. [tex]2sin x + 3cos x = 5[/tex][tex]2\frac{2t}{1+t^2} + 3\frac{1-t^2}{1+t^2} = 5[/tex]
[tex]\frac{4t + 3 - 3t^2}{1+t^2} = 5[/tex]
[tex]4t + 3 -3t^2 = 5 +5t^2[/tex]
[tex]4t + 2t^2 = 2[/tex]
[tex]2t^2 + 4t - 2 = 0[/tex]
[tex]t = 0.4142135624[/tex] or [tex]t = -2.414213562[/tex]
[tex]t = \tan\frac{x}{2}[/tex]
[tex]x = 2tan^-1 0.4142135624[/tex] or [tex]x = 2tan^-1 -2.414213562[/tex]
[tex]x = 45[/tex] or [tex]x = -135.0005034[/tex]
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I conclude my working here as I am not sure if it is correct. Is there an aleternaitve method I could use to check these final answers? And are there any more values of x for which the equation is true?Thanks well in advance,
Pavadrin