- •Functions
- •The Concept of a Function
- •Trigonometric Functions
- •Inverse Trigonometric Functions
- •Logarithmic, Exponential and Hyperbolic Functions
- •Limits and Continuity
- •Introductory Examples
- •Continuity Examples
- •Linear Function Approximations
- •Limits and Sequences
- •Properties of Continuous Functions
- •The Derivative
- •The Chain Rule
- •Higher Order Derivatives
- •Mathematical Applications
- •Antidifferentiation
- •Linear Second Order Homogeneous Differential Equations
- •Linear Non-Homogeneous Second Order Differential Equations
- •Area Approximation
- •Integration by Substitution
- •Integration by Parts
- •Logarithmic, Exponential and Hyperbolic Functions
- •The Riemann Integral
- •Volumes of Revolution
- •Arc Length and Surface Area
- •Techniques of Integration
- •Integration by formulae
- •Integration by Substitution
- •Integration by Parts
- •Trigonometric Integrals
- •Trigonometric Substitutions
- •Integration by Partial Fractions
- •Fractional Power Substitutions
- •Numerical Integration
- •Integrals over Unbounded Intervals
- •Discontinuities at End Points
- •Improper Integrals
- •Sequences
- •Monotone Sequences
- •Infinite Series
- •Series with Positive Terms
- •Alternating Series
- •Power Series
- •Taylor Polynomials and Series
- •Applications
- •Parabola
- •Ellipse
- •Hyperbola
- •Polar Coordinates
- •Graphs in Polar Coordinates
- •Areas in Polar Coordinates
- •Parametric Equations
1.3. |
INVERSE TRIGONOMETRIC FUNCTIONS |
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If f(x) = cos x, prove that |
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cos h |
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− sin x |
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f(x + h) |
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h |
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sin h |
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If f(x) = sin x, prove that |
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cos h |
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+ cos x |
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f x |
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h |
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sin h |
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If f(x) = cos x, prove that |
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x − t |
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x − t |
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cos(x − t) − |
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sin t |
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If f(x) = sin x, prove that |
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x − t |
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x − t |
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x − t |
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f(x) − f(t) |
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+ cos t |
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Prove that |
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1 − tan2 t |
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cos(2t) = |
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1 + tan2 t |
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14. Prove that if y = tan x2 , then
(a) cos x = |
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1.3Inverse Trigonometric Functions
None of the trigonometric functions are one-to-one since they are periodic. In order to define inverses, it is customary to restrict the domains in which the functions are one-to-one as follows.
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CHAPTER 1. |
FUNCTIONS |
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1. y = sin x, − |
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π |
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Its inverse function is denoted |
arcsin x, and we define y = arcsin x, |
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x ≤ 1, if and only if, x = sin y, − |
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graph
2. y = cos x, 0 ≤ x ≤ π, is one-to-one and covers the range −1 ≤ y ≤ 1. Its inverse function is denoted arccos x, and we define y = arccos x, −1 ≤ x ≤ 1, if and only if, x = cos y, 0 ≤ y ≤ π.
graph
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−∞ |
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3. y = tan x, |
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y < ∞ Its inverse function is denoted arctan x, and we define y = |
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∞ |
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arctan x, |
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< x < |
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graph
4. y = cot x, 0, x < π, is one-to-one and covers the range −∞ < y < ∞. Its inverse function is denoted arccot x, and we define y = arccot x, −∞ < x < ∞, if and only if x = cot y, 0 < y < π.
graph
1.3. INVERSE TRIGONOMETRIC FUNCTIONS |
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5. y = sec x, 0 ≤ x ≤ |
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−∞ < y ≤ −1 or 1 ≤ y < ∞. Its inverse function is denoted arcsec x, |
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arcsec x, |
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and we define |
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if, x = sec y, 0 ≤ y < |
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or |
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6. y = csc x, −2π ≤ x < 0 or 0 < x ≤ π2 , is one-to-one and covers the range −∞ < y ≤ −1 or 1 ≤ y < ∞. Its inverse is denoted arccsc x and
we define y = arccsc x, −∞ < x ≤ −1 or 1 ≤ x < ∞, if and only if, x = csc y, −2π ≤ y < 0 or 0 < y ≤ π2 .
Example 1.3.1 Show that each of the following equations is valid.
(a)arcsin x + arccos x = π2
(b)arctan x + arccot x = π2
(c)arcsec x + arccsc x = π2
To verify equation (a), we let arcsin x = θ.
graph
Then x = sin θ and cos that
π2 − θ = arccos x,
π2 − θ = x, as shown in the triangle. It follows
π2 = θ + arccos x = arcsin x + arccos x.
The equations in parts (b) and (c) are verified in a similar way.
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CHAPTER 1. FUNCTIONS |
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Example 1.3.2 If θ |
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csc θ. |
π |
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π |
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If θ is − |
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graph |
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Suppose that − |
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π |
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cos θ = √ |
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cot θ = |
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1 − x2 |
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tan θ = |
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√1 − x2 |
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sec θ = |
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and csc θ = |
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Example 1.3.3 Make the given substitutions to simplify the given radical
expression and compute all trigonometric functions of θ.
√ √
(a) 4 − x2, x = 2 sin θ (b) x2 − 9, x = 3 sec θ
(c)(4 + x2)3/2, x = 2 tan θ
(a)For part (a), sin θ = x2 and we use the given triangle:
graph
Then
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√ |
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cos θ = |
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tan θ = |
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cot θ = |
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sec θ = |
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csc θ = |
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Furthermore, 4 − x2 = 2 cos θ and the radical sign is eliminated.
1.3. INVERSE TRIGONOMETRIC FUNCTIONS |
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(b)For part (b), sec θ = x3 and we use the given triangle:
graph
Then,
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sin θ = |
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cot θ = |
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csc θ = |
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x2 − 9 |
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Furthermore, x2 − 9 = 3 tan θ and the radical sign is eliminated.
(c)For part (c), tan θ = x2 and we use the given triangle:
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Then, |
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sin θ = |
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sec θ = |
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Furthermore, x2 + 4 = 2 sec θ and hence
(4 + x)3/2 = (2 sec θ)3 = 8 sec3 θ.
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CHAPTER 1. FUNCTIONS |
Remark 2 The three substitutions given in Example 15 are very useful in calculus. In general, we use the following substitutions for the given radicals:
√ √
(a) a2 − x2, x = a sin θ (b) x2 − a2, x = a sec θ
√
(c)a2 + x2, x = a tan θ.
Exercises 1.3 |
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1. Evaluate each of the following: |
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(d)cos(2 arccos(x))
(e)sin(2 arccos(x))
2.Simplify each of the following expressions by eliminating the radical by using an appropriate trigonometric substitution.
(a) |
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√9 − x2 |
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√x2 + 2x + 2 |
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(Hint: In parts (d) and (e), complete squares first.)
3.Some famous polynomials are the so-called Chebyshev polynomials, defined by
Tn(x) = cos(n arccos x), −1 ≤ x ≤ 1, n = 0, 1, 2, . . . .
1.3. INVERSE TRIGONOMETRIC FUNCTIONS |
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(a) Prove the recurrence relation for Chebyshev polynomials:
Tn+1(x) = 2xTn(x) − Tn−1(x) for each n ≥ 1.
(b)Show that T0(x) = 1, T1(x) = x and generate T2(x), T3(x), T4(x) and T5(x) using the recurrence relation in part (a).
(c)Determine the zeros of Tn(x) and determine where Tn(x) has its absolute maximum or minimum values, n = 1, 2, 3, 4, ?.
(Hint: Let θ = arccos x, x = cos θ. Then Tn(x) = cos(nθ), Tn+1(x) = cos(nθ + θ), Tn−1(x) = cos(nθ − θ). Use the expansion formulas and then make substitutions in part (a)).
4.Show that for all integers m and n,
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(Hint: use the expansion formulas as in problem 3.) 5. Find the exact value of y in each of the following
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y = arcsin |
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y = arcsec −√2 |
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y = arccsc (−2) |
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c) y = arctan(− 3)
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i)y = arcsec (−2)
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l) y = arccot (− 3)
6. Solve the following equations for x in radians (all possible answers).
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2 sin4 x = sin2 x |
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2 cos2 x − cos x − 1 = 0 |
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sin2 x + 2 sin x + 1 = 0 |
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4 sin2 x + 4 sin x + 1 = 0 |