SM2.3 Introduction to trigonometry
You can define the six trigonometric ratios in a right triangle, evaluate them for the standard angles, use the complementary-angle relations, and prove simple identities.
Topics shown in module order.
In a right triangle, once one acute angle is fixed, the ratios between the sides are fixed too, whatever the triangle's size. Those ratios get names, sine, cosine, tangent and their reciprocals, and a small table of exact values at the standard angles. One identity generated by the Pythagoras theorem connects them all, and most of the chapter's proving questions are that identity in costume.
Work through these
From a labelled right triangle, write all six ratios for one acute angle, then all six for the other
Writing both angles' ratios from one triangle shows why sine of one equals cosine of the other: the opposite and adjacent sides trade places. The complementary-angle relations stop being a rule to learn and become something you saw happen.
Build the table of exact values for the standard angles from the two special triangles, not from memory
The half-square and the half-equilateral triangle generate every value in the table. Building it twice from scratch means an examination blank can be refilled in a minute instead of costing the question.
Given one ratio, find the others by drawing the triangle it describes
One ratio pins the triangle's shape, and Pythagoras supplies the third side. Drawing it beats juggling identities at this stage because the picture cannot lose a sign or a reciprocal.
Prove a handful of identities, starting each from the Pythagorean identity and saying which side you are transforming
Identity proofs reward a steady method: pick a side, transform it, and never move terms across the equals sign as if it were an equation. Starting from the one identity you can always derive keeps the work grounded.
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