Exponentials wearing trig costumes

Mathematics · Algebra · ages 17-18

Name ______________________ Date ____________

What is cosh 0 minus sinh 0?
Answer: ______________
Which defines sinh x in terms of exponentials?
- (e^x + e^{-x})/2
- (e^x - e^{-x})/2
- e^x - e^{-x}
- 2/(e^x - e^{-x})
Which defines sinh x in terms of exponentials?
- (e to the x plus e to the minus x) over 2
- (e to the x minus e to the minus x) over 2
- e to the x minus e to the minus x
Zoe says cosh squared x minus sinh squared x equals 1 for every real x. Is Zoe right?
Circle one: True False
Let f(x) = sinh x + cosh x. What is f'(0)?
Answer: ______________
cosh squared x minus sinh squared x equals what?
- 1
- 0
- sech squared x
Which point lies on the graph y = cosh x?
- (0, 0)
- (1, 0)
- (0, e)
- (0, 1)
Max says arsinh x = ln(x + sqrt(x squared + 1)) is defined for every real x. Is Max right?
Circle one: True False
Max says arsinh x equals ln(x plus root(x squared plus 1)) is defined for every real x. Is Max right?
Circle one: True False
Which circular identity is the counterpart of cosh squared x minus sinh squared x = 1?
- cos squared x + sin squared x = 1
- tan squared x + cot squared x = 1
- sin squared x - cos squared x = 1
- sec squared x - tan squared x = 0

Answer key

For grown-ups. Fold this page away before handing over the rest.

1 · 1. cosh 0 = 1 and sinh 0 = 0, so the difference is 1.
(e^x - e^{-x})/2 · sinh x = (e^x - e^{-x})/2. The plus version defines cosh x instead.
(e to the x minus e to the minus x) over 2 · sinh x is the half difference, while the half sum defines cosh x.
True · Zoe is right. Expanding both definitions leaves 1 after cancellation.
1 · 1. f' = cosh + sinh, giving 1 + 0 = 1 at zero.
1 · This is the hyperbolic identity, the minus sign partner of the circular plus identity.
(0, 1) · (0, 1). cosh 0 = 1, while (0, 0) belongs to sinh.
True · Max is right. Since x + sqrt(x squared + 1) stays positive for all real x, the log domain is the whole line.
True · Max is right. The log input stays positive for all real x.
cos squared x + sin squared x = 1 · cos squared x + sin squared x = 1. The sign flips from minus to plus between the hyperbolic and circular cases.