Think book answer may be wrong

Question

Hi,

Our school course book is Understanding Pure Mathematics by Sadler and Thorning and we are asked to solve cot(2x) = -2/5 for which the book gives four answers: 55.9, 145.9, 235.9, and 325.9. I calculated 2x as 68.19 and, thus, x as 34.1 and as tan(x) = -5/2 the answers must be in the second and fourth quadrant. As a result, I am able to get 145.9 (2nd quadrant) and 325.9 (4th quadrant) but can't see how Sadler and Thorning also get 55.9 and 235.9. I have tried everything and can only assume there is a mistake in the book. I sincerely hope that somehow someone can show me where I have gone wrong (as I don't want to study from a book which contains mistakes!).

Thanks,

Christian B.

Answer 1

Here is how to get those answers:

cot(2x) = -2/5

2x = cot^-1(-2/5)

2x = 111.8 + 180k, k any integer (this gives all possible values of 2x)

x = 55.9 + 90 k, k any integer (divide both sides by 2)

x= 55.9, 55.9+90, 55.9+180, 55.9+270 (let k take on values 0,1,2,3)

The above assumes using degree mode.

Answer 2

55.9 * 2 = 111.8

tan (111.8) = -2.5, so cot = -0.4

I'd say the book is right. Plotting the double angles on graph paper may make things clearer. The double angles are:

111.8, 291.8, 471.8, 651,8

Subtract from 180*n to get the equivalent angles:

180 - 111.8 = 68.2

360 - 291.8 = 68.2

540 - 471.8 = 68.2

720 - 651.8 = 68.2

PS: "equivalent", but with the opposite sign for tan and cot. eg tan(x) = - tan(180 - x)

Answer 3

For Mark H's answer: