Download The Inverse Sine, Cosine, and Tangent Functions

Section 8.1: The Inverse Sine, Cosine, and Tangent
Functions
• The function y = sin x doesn’t pass the horizontal line test, so it doesn’t have
an inverse for every real number.
But if we restrict the function to only on
π
cycle; i.e., to the interval −π
,
, the the function is one-to-one and so it
2 2
does have an inverse.
• Def: The inverse sine, also called the arcsine, is the function y = sin−1 x =
arcsin x, which is the inverse of the function x = sin y. The domain of the
inverse sine is −1 ≤ x ≤ 1 and the range is − π2 ≤ y ≤ π2 . The graph of
y = sin−1 x looks like:
• Since sin x and sin−1 x are inverses of each other, we have the following relationships:
1. sin−1 (sin x) = x, provided that − π2 ≤ x ≤ π2 .
2. sin sin−1 x = x, provided that −1 ≤ x ≤ 1.
In the first equation, if x is not between − π2 and π2 , then you first need to
figure out which quadrant x is in. If x is in quadrants I or IV, then change
x to its coterminal angle which is between − π2 and π2 . If x is in quadrant II,
change x for its reference angle. If x is in quadrant III, change x to the angle
in quadrant IV which has the same reference angle as x.
In the second equation, if x is not between −1 and 1, then the composition
is undefined.
• Def: The inverse cosine, also called the arccosine, is the function y = cos−1 x =
arccos x, which is the inverse of the function x = cos y. The domain of the
1
inverse cosine is −1 ≤ x ≤ 1 and the range is 0 ≤ y ≤ π. The graph of
y = cos−1 x looks like:
• Since cos x and cos−1 x are inverses of each other, we have the following
relationships:
1. cos−1 (cos x) = x, provided that 0 ≤ x ≤ π.
2. cos (cos−1 x) = x, provided that −1 ≤ x ≤ 1.
In the first equation, if x is not between 0 and π, then you first need to figure
out which quadrant x is in. If x is in quadrants I or II, then change x to its
coterminal angle which is between 0 and π. If x is in quadrant III, change x
to the angle in quadrant II which has the same reference angle as x. If x is
in quadrant IV, then change x for its reference angle.
In the second equation, if x is not between −1 and 1, then the composition
is undefined.
• Def: The inverse tangent, also called the arctangent, is the function y =
tan−1 x = arctan x, which is the inverse of the function x = tan y. The
domain of the inverse tangent is −∞ < x < ∞ and the range is − π2 < y < π2 .
The graph of y = tan−1 x looks like:
2
• Since tan x and tan−1 x are inverses of each other, we have the following
relationships:
1. tan−1 (tan x) = x, provided that − π2 < x < π2 .
2. tan (tan−1 x) = x, provided that −∞ < x < ∞.
In the first equation, if x is not between − π2 and π2 , then you first need to
figure out which quadrant x is in. If x is in quadrants I or IV, then change
x to its coterminal angle which is between − π2 and π2 . If x is in quadrant
II then change x to the angle in quadrant IV which has the same reference
angle as x. If x is in quadrant III, then change x for its reference angle.
• ex. Find the exact value of each expression.
√ (a) cos−1 22
√ (b) tan−1 − 3
• ex. Find the exact value, if any, of each expression.
(a) sin−1 sin 3π
5
3
(b) sin sin−1
3
10
(c) cos−1 cos − 3π
4
(d) cos [cos−1 (π)]
(e) tan−1 tan
11π
5
4
Section 8.2: The inverse Trigonometric Functions
(Continued)
• Def: The inverse secant, also called the arcsecant, is the function y = sec−1 x =
arcsec x, which is the inverse of the function x = sec
of the
y. The domain
inverse secant is (−∞, 1] ∪ [1, ∞) and the range is 0, π2 ∪ π2 , π .
• Def: The inverse cosecant, also called the arccosecant, is the function y =
csc−1 x = arccsc x, which is the inverse of the function x = csc
y.π The
domain
of the inverse cosecant is (−∞, 1] ∪ [1, ∞) and the range is − 2 , 0 ∪ 0, π2 .
• Def: The inverse cotangent, also called the arccotangent, is the function
y = cot−1 x = arccot x, which is the inverse of the function x = tan y. The
domain of the inverse tangent is −∞ < x < ∞ and the range is 0 < y < π.
• Note: The inverse of a trig function is asking what angle in the domain would
be needed to give the trig value the given value. So to find the exact value
of a trig expression involving a trig function composed with an inverse trig
function which are not inverses of each other, use the inverse trig function to
draw a right triangle and use the triangle to solve the problem.
• ex. Find the exact value of each expression.
(a) tan cos−1 − 13
(b) sec cos−1 − 43
1
(c) sin−1 cos 3π
4
(d) cot csc−1
√ 10
• ex. Write each trigonometric expression as an algebraic expression in u.
(a) cos sin−1 u
(b) tan (csc−1 u)
2
Section 8.3 (Previously Section 8.7 & 8.8):
Trigonometric Equations
• Recall that the period of sin x, cos x, csc x, & sec x is 2π and the period of
tan x & cot x is π. Thus,
θ (Degrees)
θ (Radians)
◦
sin (θ + 360 n) = sin θ
sin (θ + 2πn) = sin θ
◦
cos (θ + 360 n) = cos θ
cos (θ + 2πn) = cos θ
◦
tan (θ + 360 n) = tan θ
tan (θ + 2πn) = tan θ
◦
csc (θ + 360 n) = csc θ
csc (θ + 2πn) = csc θ
◦
sec (θ + 360 n) = sec θ
sec (θ + 2πn) = sec θ
◦
cot (θ + 360 n) = cot θ
cot (θ + 2πn) = cot θ
• ex. Solve each equation on the interval 0 ≤ θ < 2π.
(a) sin (2θ) + 1 = 0
(b) sec2 θ = 4
1
(c) 4 sin2 θ − 3 = 0
(d) cos
θ
3
−
π
4
=
1
2
• ex. Give a general formula for all the solutions. List six solutions.
(a) cos θ =
1
2
(b) cot θ = 1
(c) sin (2θ) = − 12
• ex. Solve each equation on the interval 0 ≤ θ < 2π.
(a) 2 sin2 θ − 3 sin θ + 1 = 0
2
(b) 8 − 12 sin2 θ = 4 cos2 θ
(c) 1 +
√
3 cos θ + cos (2θ) = 0
(d) sin θ −
√
3 cos θ = 2
3
Section 8.4 (Previously Section 8.3): Trigonometric
Identities
• ex. Establish each identity.
(a) tan θ cot θ − sin2 θ = cos2 θ
(b)
cos θ
cos θ−sin θ
=
1
1−tan θ
1
(c) 1 −
sin2 θ
1+cos θ
= cos θ
(d) csc θ − sin θ = cos θ cot θ
2
Section 8.5 (Previously Section 8.4): Sum and Difference
Formulas
• Theorem (Sum and Difference Formulas)
1.
2.
3.
4.
5.
sin (x + y) = sin x cos y + cos x sin y
sin (x − y) = sin x cos y − cos x sin y
cos (x + y) = cos x cos y − sin x sin y
cos (x − y) = cos x cos y + sin x sin y
tanx+tan y
tan (x + y) = 1−tan
x tan y
6. tan (x − y) =
tan x−tan y
1+tan x tan y
• ex. Find the exact value of each expression.
(a) cos 15◦
(b) tan 75◦
(c) sin 165◦
(d) sec 105◦
(e) csc
11π
12
1
(f) cot − 5π
12
• ex. Find the exact value of (a) sin (x + y), (b) cos (x + y), (c) tan (x − y)
given that
3
3π
12 3π
sin x = − , π < x <
; cos y = ,
< y < 2π
5
2
13 2
• ex. Establish each identity.
(a) sin (π + θ) = − sin θ
2
(b)
sin (x−y)
sin x cos y
= 1 − cot x tan y
• ex. Find the exact value of each expression.
(a) cos sin−1 53 − cos−1 12
(b) tan sin−1 − 12 − tan−1 43
3
Section 8.6 (Previously Section 8.5): Double-angle and
Half-angle Formulas
• Theorem (Double-angle Formulas)
1. sin (2θ) = 2 sin θ cos θ
2. cos (2θ) = cos2 θ − sin2 θ
3. cos (2θ) = 1 − 2 sin2 θ
4. cos (2θ) = 2 cos2 θ − 1
5. tan (2θ) =
2 tan θ
1−tan2 θ
• Note: Formulas 1, 2, and 5 can be obtained from the Sum Formulas from
the previous section by setting x = θ and y = θ. Formulas 3 and 4 can be
obtained from formula 2 by using the Pythagorean Identity sin2 θ+cos2 θ = 1.
In formula 3, solve the Pythagorean Identity for cos2 θ and plugging it into
formula 2. In formula 4, solve the Pythagorean Identity for sin2 θ and plugging
it into formula 2.
• From the Double-angle formulas, we can get formulas for the square of the
trig functions.
1−cos (2θ)
2
1+cos (2θ)
=
2
1−cos (2θ)
= 1+cos (2θ)
1. sin2 θ =
2. cos2 θ
3. tan2 θ
• In the previous set of formulas for the square of the trig functions, if we
replace each θ by φ2 , we get the following formulas:
φ
φ
= 1−cos
2
2
φ
cos2 φ2 = 1+cos
2
φ
tan2 φ2 = 1−cos
1+cos φ
1. sin2
2.
3.
• Theorem (Half-angle Formulas)
q
θ
θ
1. sin 2 = ± 1−cos
2
q
θ
2. cos 2θ = ± 1+cos
2
q
θ
3. tan 2θ = ± 1−cos
1+cos θ
4. tan 2θ =
5. tan 2θ =
1−cos θ
sin θ
sin θ
1+cos θ
where the + or − sign is determined by the quadrant in which the angle
lies in.
1
θ
2
• ex. Find (a) cos (2θ), (b) sin 2θ given that
3
3π
sin θ = − , π < θ <
5
2
• ex. Find the exact value of each expression.
(a) cos 15◦
(b) tan π8
• ex. Establish each identity.
(a) 2 sin (2θ) cos (2θ) = sin (4θ)
2
(b) sin (3θ) = 3 sin θ − 4 sin3 θ
• ex. Find the exact value of each expression.
(a) sin 21 cos−1 35
(b) tan 2 sin−1
6
11
3
Section 8.7 (Previously Section 8.6): Product-to-Sum
and Sum-to-Product Formulas
• Theorem (Product-to-Sum Formulas)
1. sin x sin y = 21 [cos (x − y) − cos (x + y)]
2. cos x cos y = 21 [cos (x − y) + cos (x + y)]
3. sin x cos y = 12 [sin (x + y) + sin (x − y)]
• Theorem (Sum-to-Product Formulas):
cos x−y
1. sin x + sin y = 2 sin x+y
2
2
2. sin x − sin y = 2 sin x−y
cos x+y
2
2
cos x−y
3. cos x + cos y = 2 cos x+y
2
2
4. cos x − cos y = −2 sin x+y
sin x−y
2
2
• ex. Express each product as a sum containing only sines or only cosines.
(a) sin (3θ) sin (4θ)
(b) cos (3θ) cos (2θ)
(c) sin
θ
2
cos
3θ
2
• ex. Express each sum or difference as a product of sines and/or cosines.
(a) sin 2θ + sin (4θ)
(b) cos (5θ) + cos θ
1
(c) cos
θ
2
− cos
5θ
2
• ex. Establish each identity.
(a)
sin (2θ)+sin (4θ)
cos (2θ)+cos (4θ)
= tan (3θ)
(b) sin θ [sin (3θ) + sin (5θ)] = cos θ [cos (3θ) − cos (5θ)]
2