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Question

Find the principal values of each of the following:
(i) tan-1(-3)
(ii) cos-1-12
(iii) cosec-1(-2)
(iv) cos-1-32
(v) sin-1-12
(vi) tan-113
(vii) sec-1(-2)
(viii) cot-1(-3)
(ix) sec-1(2)
(x) cosec-123

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Solution

(i) Let tan-1-3=y
Then,
tany=-3
We know that the range of the principal value branch is -π2, π 2.
Thus,
tany=-3=-tanπ3=tan-π3y=-π3-π2,π2
Hence, the principal value of tan-1-3 is -π3.

(ii) Let cos-1-12=y
Then,
cosy=-12
We know that the range of the principal value branch is 0, π.
Thus,
cosy=-12=cos3π4y=3π40,π
Hence, the principal value of cos-1-12 is 3π4.

(iii) ​ Let cosec-1-2=y
Then,
cosecy=-2
We know that the range of the principal value branch is -π2, π2-0.
Thus,
cosecy=-2=cosec-π4y=-π4-π2,π2, y0
Hence, the principal value of cosec-1-2 is -π4.

(iv) ​ ​ Let cos-1-32=y
Then,
cosy=-32
We know that the range of the principal value branch is 0,π.
Thus,
cosy=-32=cos5π6y=5π60,π
Hence, the principal value of cos-1-32 is 5π6.

(v) Let sin-1-12=y
Then,
siny=-12
We know that the range of the principal value branch is -π2,π2.
Thus,
siny=-12=sin-π6y=-π6-π2,π2
Hence, the principal value of sin-1-12 is -π6.

(vi) Let tan-113=y
Then,
tany=13
We know that the range of the principal value branch is -π2,π2.
Thus,
tany=13=tanπ6y=π6-π2,π2
Hence, the principal value of tan-113 is π6.

(vii) Let sec-1-2=y
Then,
secy=-2
We know that the range of the principal value branch is 0,π-π2.
Thus,
secy=-2=sec3π4y=3π40,π, yπ2
Hence, the principal value of sec-1-2 is 3π4.

(viii) Let cot-1-3=y
Then,
coty=-3
We know that the range of the principal value branch is 0,π.
Thus,
coty=-3=cot5π6y=5π60,π
Hence, the principal value of cot-1-3 is 5π6.

(ix) Let sec-12=y
Then,
secy=2
We know that the range of the principal value branch is 0,π-π2.
Thus,
secy=2=secπ3y=π30, π, yπ2
Hence, the principal value of sec-12 is π3.

(x) Let cosec-123=y
Then,
cosecy=23
We know that the range of the principal value branch is -π2, π2-0.
Thus,
cosecy=23=cosecπ3y=π3-π2, π2, y0
Hence, the principal value of cosec-123 is π3.

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