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Solution 4.4:3c

From Förberedande kurs i matematik 1

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If we consider the entire expression <math>x + 40^{\circ}</math> as an unknown, we have a basic trigonometric equation and can, with the aid of the unit circle, see that there are two solutions to the equation for <math>0^{\circ}\le x+40^{\circ}\le 360^{\circ}</math> namely <math>x+40^{\circ} = 65^{\circ}</math> and the symmetric solution <math>x + 40^{\circ} = 180^{\circ} - 65^{\circ} = 115^{\circ}\,</math>.
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[[Image:4_4_3_c.gif|center]]
[[Image:4_4_3_c.gif|center]]
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It is then easy to set up the general solution by adding multiples of <math>360^{\circ}\,</math>,
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{{Displayed math||<math>x + 40^{\circ} = 65^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x + 40^{\circ} = 115^{\circ} + n\cdot 360^{\circ}</math>}}
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for all integers ''n'', which gives
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{{Displayed math||<math>x = 25^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x=75^{\circ} + n\cdot 360^{\circ}\,\textrm{.}</math>}}

Current revision

If we consider the entire expression x+40 as an unknown, we have a basic trigonometric equation and can, with the aid of the unit circle, see that there are two solutions to the equation for 0x+40360 namely x+40=65 and the symmetric solution x+40=180−65=115.

It is then easy to set up the general solution by adding multiples of 360,

x+40=65+n360andx+40=115+n360

for all integers n, which gives

x=25+n360andx=75+n360.
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