There are no short cuts when it comes to learning trigonometry. To manipulate the sine and cosine formulae and to get a handle on all those angles, the student must concentrate, and concentrate hard.
It's not so bad at GCSE. But anyone who tackles advanced trigonometry must be prepared to put in the hours.
And that's just what Sydney Chadwick does. On a winter's evening, when the rest of the world is watching telly, he likes nothing better than to sit down in front of the fire and master complex calculations. "I enjoy it," he says. "The mental activity keeps my mind alert."
And alertness is not to be taken for granted at his time of life. For Sydney Chadwick is 97 years old. Mr Chadwick, who lives and studies in Uttoxeter in Staffordshire, is one of a growing band of elderly learners - people who, in the words of Malcolm Wicks, minister for lifelong learning, "are an inspiration, not just to people in their own age group, but to all of us".
When Mr Wicks launched the Learning in Later Life Campaign 2000 to find England's oldest and "most inspiring" learners, 155 people came forward. They were learning everything from creative writing and T'ai Chi to flamenco and Sanscrit.
After 25 years of attending art classes, Fred Moore has recently taken an interest in current affairs and computers. At 107 years old, he has the title of England's oldest learner.
In all, the campaign awarded 21 titles. Recipients included Emily May Butterfield from Taunton, Somerset (England's most inspiring learner), Arthur Hodges from Dagenham, Essex (the eastern region's most inspiring learner), and Hetty Kemp (joint winner of London's most inspiring learner).
At 97, Emily May Butterfield is not only researching Captain Cook, but she's doing it with a computer, while 90-year-old Arthur Hodges has also studied computer science, word-processing, accounting, music theory, Russian, Greek, Dutch, German, Spanish and Italian - all since retiring at the age of 70.
"Learning in later life restores the status and dignity often lost on retirement," says Mr Hodges. "Mental activity also keeps you fit, prolongs active life and keeps loneliness and depression at bay."
Like Arthur Hodges, Hetty Kemp, 86, is a living advertisement for the benefits of learning in later life. In the past 10 years she has studied cookery, fitness, flamenco and belly-dancing. She learned to drive at 75, and now does ballet. "I always wanted to dance as a child," she says, "but my parents couldn't afford the classes. Learning has kept me independent, active and mobile. It keeps me in touch with enthusiastic and motivated people of all ages and walks of life and has made my old age my new age."
According to Professor Alan Tuckett, director of the National Institute for Adult Continuing Education (NIACE), many of the award winners took up classes on the advice of their GP. "What stimulates doctors to encourage people to join in a learning activity rather than giving them a bunch of pills?" he asks. Perhaps, he says, it is the suspicion that learning can keep people healthy.
In a recent NIACE survey, 87 per cent of respondents said learning made them feel physically better or helped them cope with ill health; 89 per cent reported emotional or mental health benefits. " 'Use it or lose it' is a generally understood nostrum," says Professor Tuckett. "And we are all familiar with the idea that we accelerate morbidity by not doing things."
Some researchers have gone so far as to suggest that learning in old age can delay the onset of diseases such as Alzheimer's. But while Alex Withnall, a lecturer in gerontology at Warwick University, is sceptical about some of the more extravagant claims, she believes there is increasing evidence that mental stimulation can help to improve later-life capabilities.
"Some evidence from the United States suggests that mental training in later life can boost intellectual power, ward off some of the degenerative diseases, perhaps assist in maintaining mental function and reverse memory decline," she says. "My own feeling is that, if learning does improve health, it's less due to the actual learning than the social interaction."
While researchers may argue about precise causes and effects, those who work with the elderly have no doubt as to the benefits of learning in later life. Jim Soulsby, programme development officer with a NIACE initiative called Older and Bolder, terms it a "social solution".
He says: "Part of the difficulty is that ageing is seen as a medical model, so there's a tendency for the medical fraternity to dominate the debate and to look for medical solutions. But a lot of us try to work on social solutions."
Not that he is comfortable talking about solutions. "Many people see ageing as a problem," he says. "But there's a lovely expression - 'Growing older is the only way we can live longer'. The emphasis is on growing, not degenerating. It makes you stop and think.
"You can see 35-year-olds who are old and who have given up, and you can see 85-year-olds who will go on forever. Should we even be looking for solutions?" But if ageing itself is not a problem, the exclusion of older people from the education system most certainly is, says Mr Soulsby. "In the past, people over 50 have not been present in education. They have been leaving work in droves, family life is behind them and there is a sense of being excluded from society. We're trying to reverse that."
Older and Bolder has been tackling the problem on several fronts - by collecting and disseminating information, establishing standards of good practice and encouraging existing providers to offer more and better learning opportunities for old people.
"We are also trying to overcome misconceptions about ability to learn - the idea that you can't teach an old dog new tricks," says Jim Soulsby. "There's something about older people and new technology in the newspapers nearly every day. But some attitude-changing needs to take place. Sometimes perceptions are based on people's early school experience."
Unhappy memories of schooldays frequently deter older people from taking up learning opportunities, and according to Alex Withnall it is still the people who had the most education initially who are most likely to start studying again in later life. "But we are seeing people who didn't have those opportunities realising that it's not too late," she says.
Ms Withnall is running a series of focus groups to explore older people's attitudes to learning, and it was a participant in one of them who coined the phrase "indulgent learning".
Alex Withnall says: "Indulgent learning is about people being able to learn things they've always wanted to instead of learning what other people think they should learn. We've found people picking up literature that they perhaps touched on at school - and hated, and people learning languages and getting interested in all kinds of things that wouldn't have been on their school curriculum or that they wouldn't have learned in their work."
Ms Withnall wants people in general - and the Government in particular - to realise that lifelong learning really does mean that. "It's not just for people who are in work, which has been very much the thrust of the Government's drive," she says. "The whole thing has been oriented towards vocational training and education. But I think they're realising now that they have to include people who are post-work.
"At the moment there is a huge variation in what's available, and in the cost of it. The cost can be a problem for older people. And then there is timing, transport and so on.
"One London borough had a scheme that involved matching housebound older people with volunteers who could teach them guitar or a language in their home. I think increasingly residential homes will provide learning activities rather than let people sit staring blankly at the television all day."
Ms Withnall believes provision is bound to improve as the generation that benefited from post-war improvements in education reaches old age. "As people of my generation - those now in their fifties - get into old age, they're not going to put up with the kind of conditions 80-year-olds are putting up with now," she says. "Up-coming generations will make their mark because they are generally better educated, and I think it will be an issue in the next general election because we are an older electorate.
"I'm not saying everyone should be dragooned into going to classes. But the opportunity should be there, particularly in view of the growing evidence that continued mental stimulation could have beneficial effects."
For details of the Learning in Later Life campaign, contact Laurel Hughes at NIACE. Tel: 0116 204 4227; or visit www.niace.org.uk
While doctors are increasingly prescribing adult education to the elderly, many of those who might benefit from the therapeutic effects of learning are too frail and depressed to take part.
But a technique known as reminiscence therapy is changing all that by harnessing the power of long-term memory.
Margaret Plummer, a leading exponent of reminiscence therapy, describes it as "taking people back to the past to bring them into the present".
She co-ordinates reminiscence work for Norfolk County Council, which has invested heavily in the technique and which employs 12 reminiscence workers and three trainers.
"People who can't access their short-term memories still have their long-term memories in the back of their brain," says Ms Plummer. "But they can't access them without some sort of trigger."
To set the process in motion, workers use "memory boxes" - collections of objects that would have been commonplace half a century ago. The sight, sound, smell and feel of these objects - anything from corsets to aniseed balls - can have an extraordinary effect on someone suffering from depression or dementia who is disoriented and paralysed by the present.
"Imagine someone in a residential home," says Ms Plummer. "Recalling their past successes and triumphs and the life they once led restores their sense of self-worth. The effects are amazing.
"It's then possible to introduce them to other learning opportunities. In fact we never use it for its own sake but always to lead into other things."
As well as learning, old people in Norfolk are encouraged to teach care workers about the past. "Getting them to teach us builds their confidence," says Ms Plummer. "We are the ignorant ones. They are the ones with the skills and knowledge."