Section 8.1: 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
cycle; i.e., to the interval [ −π
, ] π
2 2 , the the function is one-to-one and so it
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 − π ≤ y ≤ π . The graph of
2
2
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 − π 2 ≤ x ≤ π 2 .
2. sin ( sin −1 x ) = x, provided that −1 ≤ x ≤ 1.
In the first equation, if x is not between − π and π , then you first need to
2 2
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 − π and π . If x is in quadrant II,
2 2
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 − π < y < π.
2 2
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 − π 2 < x < π 2 .
2. tan (tan −1 x) = x, provided that −∞ < x < ∞.
In the first equation, if x is not between − π and π , then you first need to
2 2
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 − π and π . If x is in quadrant
2 2
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.
( √2 )
(a) cos −1
2
(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 y. The domain of the
inverse secant is (−∞, 1] ∪ [1, ∞) and the range is [ 0, π 2
)
∪
( π
2 , 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. The domain
of the inverse cosecant is (−∞, 1] ∪ [1, ∞) and the range is [ − π 2 , 0) ∪ ( 0, π 2
]
.
• 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 − 1 3
(b) sec [ cos −1 ( − 3 4
)]
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)
sin (θ + 360 ◦ n) = sin θ
cos (θ + 360 ◦ n) = cos θ
tan (θ + 360 ◦ n) = tan θ
csc (θ + 360 ◦ n) = csc θ
sec (θ + 360 ◦ n) = sec θ
cot (θ + 360 ◦ n) = cot θ
θ (Radians)
sin (θ + 2πn) = sin θ
cos (θ + 2πn) = cos θ
tan (θ + 2πn) = tan θ
csc (θ + 2πn) = csc θ
sec (θ + 2πn) = sec θ
cot (θ + 2πn) = cot θ
• ex. Solve each equation on the interval 0 ≤ θ < 2π.
(a) sin (2θ) + 1 = 0
(b) sec 2 θ = 4
1

(c) 4 sin 2 θ − 3 = 0
(d) cos ( θ
− ) π
3 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) = − 1 2
• ex. Solve each equation on the interval 0 ≤ θ < 2π.
(a) 2 sin 2 θ − 3 sin θ + 1 = 0
2

(b) 8 − 12 sin 2 θ = 4 cos 2 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 − sin 2 θ = cos 2 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. sin (x + y) = sin x cos y + cos x sin y
2. sin (x − y) = sin x cos y − cos x sin y
3. cos (x + y) = cos x cos y − sin x sin y
4. cos (x − y) = cos x cos y + sin x sin y
5. tan (x + y) = tanx+tan y
6. tan (x − y) =
1−tan x tan 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
sin x = − 3 5 , π < x < 3π 2
; cos y =
12
13 , 3π
2 < y < 2π
• 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 3 − )
5 cos−1 1 2
(b) tan [ sin ( ]
−1 − 2) 1 − tan
−1 3
4
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) = cos 2 θ − sin 2 θ
3. cos (2θ) = 1 − 2 sin 2 θ
4. cos (2θ) = 2 cos 2 θ − 1
5. tan (2θ) = 2 tan θ
1−tan 2 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 sin 2 θ+cos 2 θ = 1.
In formula 3, solve the Pythagorean Identity for cos 2 θ and plugging it into
formula 2. In formula 4, solve the Pythagorean Identity for sin 2 θ and plugging
it into formula 2.
• From the Double-angle formulas, we can get formulas for the square of the
trig functions.
1. sin 2 θ =
2. cos 2 θ =
3. tan 2 θ =
1−cos (2θ)
2
1+cos (2θ)
2
1−cos (2θ)
1+cos (2θ)
• In the previous set of formulas for the square of the trig functions, if we
replace each θ by φ , we get the following formulas:
2
1. sin 2 φ 2 = 1−cos φ
2
2. cos 2 φ 2 = 1+cos φ
2
3. tan 2 φ 2 = 1−cos φ
1+cos φ
• Theorem (Half-angle Formulas)
1. sin θ 2 = ± √
1−cos θ
2
2. cos θ 2 = ± √
1+cos θ
2
3. tan θ 2 = ± √
1−cos θ
1+cos θ
4. tan θ 2 = 1−cos θ
sin θ
5. tan θ 2 = sin θ
1+cos θ
where the + or − sign is determined by the quadrant in which the angle θ 2
lies in.
1

• ex. Find (a) cos (2θ), (b) sin θ given that
2
sin θ = − 3 5 , π < θ < 3π 2
• ex. Find the exact value of each expression.
(a) cos 15 ◦
(b) tan π 8
• ex. Establish each identity.
(a) 2 sin (2θ) cos (2θ) = sin (4θ)
2

(b) sin (3θ) = 3 sin θ − 4 sin 3 θ
• ex. Find the exact value of each expression.
(a) sin ( )
1
2 cos−1 3 5
(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 = 1 [cos (x − y) − cos (x + y)]
2
2. cos x cos y = 1 [cos (x − y) + cos (x + y)]
2
3. sin x cos y = 1 [sin (x + y) + sin (x − y)]
2
• Theorem (Sum-to-Product Formulas):
1. sin x + sin y = 2 sin x+y cos x−y
2 2
2. sin x − sin y = 2 sin x−y cos x+y
2 2
3. cos x + cos y = 2 cos x+y 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