Section 8.1: The Inverse Sine, Cosine, and TangentFunctions

• The function y = sinx doesn’t pass the horizontal line test, so it doesn’t havean inverse for every real number. But if we restrict the function to only oncycle; i.e., to the interval

[−π2, π2

], 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 =arcsinx, which is the inverse of the function x = sin y. The domain of theinverse sine is −1 ≤ x ≤ 1 and the range is −π

2≤ y ≤ π

2. 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 rela-tionships:

1. sin−1 (sinx) = x, provided that −π2≤ x ≤ π

2.

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 changex to its coterminal angle which is between −π

2and π

2. If x is in quadrant II,

change x for its reference angle. If x is in quadrant III, change x to the anglein quadrant IV which has the same reference angle as x.In the second equation, if x is not between −1 and 1, then the compositionis undefined.

• Def: The inverse cosine, also called the arccosine, is the function y = cos−1 x =arccosx, 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 ofy = cos−1 x looks like:

• Since cos x and cos−1 x are inverses of each other, we have the followingrelationships:

1. cos−1 (cosx) = 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 figureout which quadrant x is in. If x is in quadrants I or II, then change x to itscoterminal angle which is between 0 and π. If x is in quadrant III, change xto the angle in quadrant II which has the same reference angle as x. If x isin quadrant IV, then change x for its reference angle.In the second equation, if x is not between −1 and 1, then the compositionis undefined.

• Def: The inverse tangent, also called the arctangent, is the function y =tan−1 x = arctanx, which is the inverse of the function x = tan y. Thedomain of the inverse tangent is −∞ < x <∞ and the range is −π

2< y < π

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 followingrelationships:

1. tan−1 (tanx) = x, provided that −π2< x < π

2.

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 changex to its coterminal angle which is between −π

2and π

2. If x is in quadrant

II then change x to the angle in quadrant IV which has the same referenceangle 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

(3π5

)]

3

(b) sin[sin−1

(310

)]

(c) cos−1[cos

(−3π

4

)]

(d) cos [cos−1 (π)]

(e) tan−1[tan

(11π5

)]

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 theinverse secant is (−∞, 1] ∪ [1,∞) and the range is

[0, π

2

)∪(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. The domainof 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 functiony = cot−1 x = arccot x, which is the inverse of the function x = tan y. Thedomain 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 wouldbe needed to give the trig value the given value. So to find the exact valueof a trig expression involving a trig function composed with an inverse trigfunction which are not inverses of each other, use the inverse trig function todraw 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 sinx, cosx, cscx, & secx is 2π and the period oftanx & cotx 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(θ3− π

4

)= 1

2

• ex. Give a general formula for all the solutions. List six solutions.

(a) cos θ = 12

(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): TrigonometricIdentities

• ex. Establish each identity.

(a) tan θ cot θ − sin2 θ = cos2 θ

(b) cos θcos θ−sin θ

= 11−tan θ

1

(c) 1− sin2 θ1+cos θ

= cos θ

(d) csc θ − sin θ = cos θ cot θ

2

Section 8.5 (Previously Section 8.4): Sum and DifferenceFormulas

• Theorem (Sum and Difference Formulas)

1. sin (x+ y) = sinx cos y + cosx sin y

2. sin (x− y) = sinx cos y − cosx sin y

3. cos (x+ y) = cos x cos y − sinx sin y

4. cos (x− y) = cos x cos y + sinx sin y

5. tan (x+ y) = tanx+tan y1−tanx tan y

6. tan (x− y) = tanx−tan y1+tanx tan y

• ex. Find the exact value of each expression.

(a) cos 15◦

(b) tan 75◦

(c) sin 165◦

(d) sec 105◦

(e) csc(11π12

)

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

sinx = −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)sinx cos y

= 1 − cotx tan y

• ex. Find the exact value of each expression.

(a) cos(sin−1 3

5− cos−1 1

2

)

(b) tan[sin−1

(−1

2

)− tan−1 3

4

]

3

Section 8.6 (Previously Section 8.5): Double-angle andHalf-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 θ1−tan2 θ

• Note: Formulas 1, 2, and 5 can be obtained from the Sum Formulas fromthe previous section by setting x = θ and y = θ. Formulas 3 and 4 can beobtained from formula 2 by using the Pythagorean Identity sin2 θ+cos2 θ = 1.In formula 3, solve the Pythagorean Identity for cos2θ and plugging it intoformula 2. In formula 4, solve the Pythagorean Identity for sin2 θ and pluggingit into formula 2.

• From the Double-angle formulas, we can get formulas for the square of thetrig functions.

1. sin2 θ = 1−cos (2θ)2

2. cos2 θ = 1+cos (2θ)2

3. tan2 θ = 1−cos (2θ)1+cos (2θ)

• In the previous set of formulas for the square of the trig functions, if wereplace each θ by φ

2, we get the following formulas:

1. sin2 φ2= 1−cosφ

2

2. cos2 φ2= 1+cosφ

2

3. tan2 φ2= 1−cosφ

1+cosφ

• Theorem (Half-angle Formulas)

1. sin θ2= ±

√1−cos θ

2

2. cos θ2= ±

√1+cos θ

2

3. tan θ2= ±

√1−cos θ1+cos θ

4. tan θ2= 1−cos θ

sin θ

5. tan θ2= sin θ

1+cos θ

where the + or − sign is determined by the quadrant in which the angle θ2

lies in.

1

• ex. Find (a) cos (2θ), (b) sin θ2given that

sin θ = −3

5, π < θ <

3π

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(12cos−1 3

5

)

(b) tan(2 sin−1 6

11

)

3

Section 8.7 (Previously Section 8.6): Product-to-Sumand Sum-to-Product Formulas

• Theorem (Product-to-Sum Formulas)

1. sinx sin y = 12

[cos (x− y) − cos (x+ y)]

2. cosx cos y = 12

[cos (x− y) + cos (x+ y)]

3. sinx cos y = 12

[sin (x+ y) + sin (x− y)]

• Theorem (Sum-to-Product Formulas):

1. sinx+ sin y = 2 sin x+y2

cos x−y2

2. sinx− sin y = 2 sin x−y2

cos x+y2

3. cosx+ cos y = 2 cos x+y2

cos x−y2

4. cosx− cos y = −2 sin x+y2

sin x−y2

• 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(θ2

)cos

(3θ2

)

• 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(θ2

)− cos

(5θ2

)

• 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