6.7.2 Hyperbolic Identities

Question 1

Hyperbolic Identities

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Which of the following is not equivalent to

3 (1 + sinh^2 x)?

Answer 3

3+3(e^2x+2+e^−2x)/4

Question 4

Which of the following is not an identity?

Answer 4

e^2x/2+e^−2x/2=2sinh^2x

Question 5

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

Answer 5

e^2x+2+e^−2x/4

Question 6

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

Answer 6

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

Question 7

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

Answer 7

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.

Question 8

Expand 2 sinh x cosh x.

Answer 8

2sinhxcoshx=e^2x−e^−2x/2=sinh2x

Question 9

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

Answer 9

csch2 x + 1

Question 10

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

Answer 10

1 − sech2 x

Question 11

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

Answer 11

2 sinh2 x

Question 12

Expand (e^x−e^−x/2)^2

Answer 12

(1/4)(e^2x−2+e^−2x)