How should I teach students the trigonometric ratios who are indeed studying trigonometry for the first time in their life? The course has said that there are six ratios based on sides in a triangle and that each ratios is given special names and that they need to memorize $\sin(\text{ref. angle})=p/h$ etc. Now They ask me what sine, cosine and tangents are? Now, what do I say them in order to make them understand what sine, cosine and tangents are?
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$\begingroup$ Something I've done is to put 45-45-90 and 30-60-90 triangles on the board with the various rules for working with them. Then I put a general $\phi$-$\theta$-90 triangle next to the other two and write the corresponding rules using trig functions, saying things like how they represent the "numbers that work" for the particular triangle. Sometimes I'd put a 20-70-90 triangle up before the general one, and give the rules using previously looked-up decimal approximations for the appropriate values (telling them I looked them up, so the numbers were not something I had memorized). $\endgroup$– Dave L RenfroApr 30, 2014 at 16:39
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As anything, show them what they are useful for. Perhaps the typical problem of computing the height of a building from distance and angle, and work from there. An understandable sort of application is surveying, setting up a base line, measure angles and compute the triangles.
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$\begingroup$ Two cents: applications in physics (e.g., work as the scalar product of displacement and force. Another one: I guess that the problem "given the triangle with sides 3, 4, 5, find its angles" might be interesting in itself for (more curious) students. $\endgroup$– mborkApr 30, 2014 at 8:47