ESSENTIAL TRIGONOMETRY (II): INVERSE TRIG FUNCTIONS
To avoid confusion with the reciprocal, write ’arctrg x’ instead of ’trg−1 x’
h π πi
= arcsin x ⇐⇒ sin θ = x where θ ∈ − ,
= QI ∪ QIV
2 2
= arccos x ⇐⇒ cos θ = x where θ ∈ [0, π] = QI ∪ QII
π π
⊂ QI ∪ QIV
= arctan x ⇐⇒ tan θ = x where θ ∈ − ,
2 2
h π πi
\ {0} ⊂ QI ∪ QIV
= arccsc x ⇐⇒ csc θ = x where θ ∈ − ,
2 2
nπo
= arcsec x ⇐⇒ sec θ = x where θ ∈ [0, π] \
⊂ QI ∪ QII
2
= arccot x ⇐⇒ cot θ = x where θ ∈ (0, π) ⊂ QI ∪ QII
INVERSE TRIG FUNCTIONS:
• θ
• θ
• θ
• θ
• θ
• θ
EXAMPLE: Find EXACTLY:
(a) θ = arccos
1
2
⇐⇒ cos θ =
(a) arccos
1
2
1
2
(b) arctan(−1)
√
(c) arccsc(− 2)
where θ ∈ [0, π] =⇒ θ = 60◦ =
(b) θ = arctan(−1) ⇐⇒ tan θ = −1 where θ ∈ − π2 ,
π
2
π
3
=⇒ θ = −45◦ = −
π
4
√
√
π
(c) θ = arccsc (− 2) ⇐⇒ csc θ = − 2 where θ ∈ − π2 , π2 \{0} =⇒ sin θ = − √12 where θ ∈ − π2 , π2 \{0} =⇒ θ = −
4
(b) arcsec − 17
(c) arccsc 0
π π
(a) θ = arcsin(−3.5) ⇐⇒ sin θ = −3.5 where θ ∈ − 2 , 2 , but −3.5 6∈ Rng(sin θ). Thus, θ ∈ ∅ =⇒ arcsin(−3.5) is undefined
π
1
1
1
is undefined
(b) θ = arcsec − 7 ⇐⇒ sec θ = − 7 where θ ∈ [0, π]\ 2 =⇒ cos θ = −7 6∈ Rng(cos θ) =⇒ arcsec −
7
EXAMPLE: Find EXACTLY:
(a) arcsin(−3.5)
(c) θ = arccsc 0 ⇐⇒ csc θ = 0 where θ ∈ − π2 , π2 \ {0} =⇒
EXAMPLE: Find EXACTLY:
(a) tan arcsin − √12
(a) θ = arcsin − √12 =⇒ sin θ = − √12
1
= 0 =⇒ 1 = 0 =⇒ arccsc 0 is undefined
sin θ
(b) cos arcsin
+ arccos − 25
(c) csc 2 arccot − 21
(
)
π π
y = −1
where θ ∈ − 2 , 2 =⇒ θ ∈ QIV =⇒
=⇒ x = 1
√
r= 2
1
4
y
−1
= tan θ := =
= −1
Hence, tan arcsin − √12
x
1
(
)
√
π π
yA = 1
1
1
(b) A = arcsin 4 =⇒ sin A = 4 where A ∈ − 2 , 2 =⇒ A ∈ QI =⇒
=⇒ xA = 15
rA = 4
(
)
√
xB = −2
2
2
B = arccos − 5 ⇐⇒ cos B = − 5 where B ∈ [0, π] =⇒ B ∈ QII =⇒
=⇒ yB = 21
rB = 5
√
√
√ 1 √21 xA
xB
yA
yB
2 15 + 21
15
−2
=⇒ cos(A+B) = cos A cos B−sin A sin B :=
−
=
−
=
−
4
5
4
5
rA
rB
rA
rB
20
(
)
√
x = −1
(c) θ = arccot − 12 =⇒ cot θ = − 21 where θ ∈ (0, π) =⇒ θ ∈ QII =⇒
=⇒ r = 5
y=2
1
1
1
1
5
= = −
Hence, csc 2 arccot − 12
= csc(2θ) =
=
:=
2
−1
sin(2θ)
2 sin θ cos θ
4
2 yr xr
√
√
2
5
5
c
2012
Josh Engwer – Revised September 3, 2012
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