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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
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
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
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
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