Expert: Sherman D. - 4/6/2009

Question
Problem 1.A.)

Solve equations below on [0, 2 Pi)

5csc(theta)= -6

Problem 1.B.)

tan^2(theta)+sec(theta)=1



Find the general solutions for the equations below


Problem 2.A.)

tan 3(theta)= -1


Problem 2.B.)

cos2(theta)=-cos(theta)

Answer
if by these you mean, and i'm using A to stand for theta.

Problem 1.A.)

Solve equations below on [0,2Pi)

5csc(A)= -6
csc(A) = -6/5
1/sin(A) = -6/5
sin(A) = -5/6
A = 303.56° or about 1.686pi , 236.44° or about 1.314pi

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Problem 1.B.)
tan(A)^2 + sec(A) = 1

tan(A)^2 = sec(A)^2 - 1

sec(A)^2 - 1 + sec(A) = 1
sec(A)^2 + sec(A) - 2 = 0

now just think of this like x^2 + x - 2 = 0

(sec(A) + 2)(sec(A) - 1) = 0

sec(A) + 2 = 0
sec(A) = -2
1/cos(A) = -2
cos(A) = -1/2
A = 120° or (2pi/3) , 240° or (4pi/3)

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Find the general solutions for the equations below

Problem 2.A.)
tan(3A)= -1
3A = 315° or (7pi/4), 135° or (3pi/4)
A = 105° or (7pi/12), 45° or (pi/4)

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Problem 2.B.)
cos(2A) = -cos(A)
2cos(A)^2 - 1 = -cos(A)
2cos(A)^2 + cos(A) - 1 = 0
(2cos(A) - 1)(cos(A) + 1) = 0

2cos(A) - 1 = 0
2cos(A) = 1
cos(A) = (1/2)
A = 60° or (pi/3), 300° or (5pi/3)

cos(A) + 1 = 0
cos(A) = -1
A = 180° or pi

ANS : (pi/3), (5pi/3), pi