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AP Calculus AB: 6.7.2 Hyperbolic Identities

Mathematics12 CardsCreated 3 months ago

This section focuses on verifying and understanding identities involving hyperbolic functions like sinh, cosh, and tanh. These identities are similar in form to trigonometric identities but are rooted in the geometry of hyperbolas rather than circles.

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Hyperbolic Identities

  • When verifying a hyperbolic identity, use the definitions of the hyperbolic functions.

  • The hyperbolic identities mirror many of the trigonometric identities.

  • The hyperbolic functions are derived from a hyperbola like, the trig functions are derived from a circle.

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Key Terms

Term
Definition

Hyperbolic Identities

  • When verifying a hyperbolic identity, use the definitions of the hyperbolic functions.

  • The hyperbolic identities mirror many...

note

  • Verify this hyperbolic identity by substituting the defining expressions of cosh x and sinh x.

  • Since the numerators are bino...

Which of the following is not equivalent to

3 (1 + sinh^2 x)?

3+3(e^2x+2+e^−2x)/4

Which of the following is not an identity?

e^2x/2+e^−2x/2=2sinh^2x

You know that cosh x=e^x+e^−x/2.
Which of the following is the expansion of
cosh2 x ?

e^2x+2+e^−2x/4

Which of the following statements related to hyperbolic functions is not correct?

cosh2 x − sinh2 x = cos2 x − sin2 x for all x.

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AP Calculus AB: 6.7.2 Hyperbolic Identities
TermDefinition

Hyperbolic Identities

  • When verifying a hyperbolic identity, use the definitions of the hyperbolic functions.

  • The hyperbolic identities mirror many of the trigonometric identities.

  • The hyperbolic functions are derived from a hyperbola like, the trig functions are derived from a circle.

note

  • Verify this hyperbolic identity by substituting the defining expressions of cosh x and sinh x.

  • Since the numerators are binomials, you need to square them using FOIL or the distributive property. A classic mistake is to square just the terms of each binomial.

  • After combining terms and canceling, you can verify that the left-hand side of the identity equals the right-hand side.

  • On the left, substituting X and Y for cos 2 x and sin 2 x illustrates their relationship to a circle.

  • Similarly, on the right, substituting X and Y for cosh 2 x and sinh 2 x illustrates their relationship to a hyperbola. Hence, this is where the name hyperbolic function comes from.

Which of the following is not equivalent to

3 (1 + sinh^2 x)?

3+3(e^2x+2+e^−2x)/4

Which of the following is not an identity?

e^2x/2+e^−2x/2=2sinh^2x

You know that cosh x=e^x+e^−x/2.
Which of the following is the expansion of
cosh2 x ?

e^2x+2+e^−2x/4

Which of the following statements related to hyperbolic functions is not correct?

cosh2 x − sinh2 x = cos2 x − sin2 x for all x.

Use x and y to represent cosh t and sinh t, respectively.
Which of these graphs represents
cosh^2 t − sinh^2 t = 1?

Let x = cosh t and y = sinh t.

cosh2 t − sinh2 t = 1

x 2 − y 2 = 1

For x = 1, y = 0.

For x = −1, y = 0.

For −1 < x < 1, there are no corresponding y-values. The graph is a hyperbola.

Expand 2 sinh x cosh x.

2sinhxcoshx=e^2x−e^−2x/2=sinh2x

You know that cosh2 x − sinh2 x = 1. Using this identity, which of the following is equivalent to coth2 x ?

csch2 x + 1

You know that cosh2 x − sinh2 x = 1. Using this identity, which of the following is equivalent to tanh2 x ?

1 − sech2 x

Which of the following is equivalent to e^2x+2+e^−2x/2 −2?

2 sinh2 x

Expand (e^x−e^−x/2)^2

(1/4)(e^2x−2+e^−2x)