Hint
Use the real definitions of the hyperbolic functions. Multiply the exact sinh and cosh values and double the result.
Worked solution
(a)
\(\displaystyle 2\sinh u\cosh u=\frac{9999}{200}\)
Answer: \(\displaystyle \frac{9999}{200}\).
Answer: \(\displaystyle \frac{9999}{200}\)
(b)
\(\displaystyle \cosh^2u+\sinh^2u=\frac{10001}{200}\)
Answer: \(\displaystyle \frac{10001}{200}\).
Answer: \(\displaystyle \frac{10001}{200}\)
(c)
\(\displaystyle \frac{\sinh (2u)}{\cosh (2u)}= \frac{9999}{10001}\)
Answer: \(\displaystyle \frac{9999}{10001}\).
Answer: \(\displaystyle \frac{9999}{10001}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Multiply the exact sinh and cosh values and double the result.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Cosh 2u adds the two squares.
(b) · 2 marks
- 1 mark: Add the two squared values.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: The denominator for tanh 2u is \(\displaystyle 1+\tanh ^{2}u\).
(c) · 2 marks
- 1 mark: Divide the double-angle sinh by cosh.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Cosh 2u adds the two squares.
Total: 6 marks.