Derive and use trig ratios for 30-60-90 and 45-45-90 special right triangles.
Derive separate exact sine, cosine, and tangent values at45°
Problem
Use a \(45-45-90\) triangle with side ratio \(1:1:\sqrt{2}\) to derive \(\sin(45°)\), \(\cos(45°)\), and \(\tan(45°)\) in that order, rationalizing radical denominators.
A unit forty-five-forty-five-ninety triangle supplies all three exact trig values from the same side roles. We’ll form opposite-over-hypotenuse, adjacent-over-hypotenuse, and opposite-over-adjacent, rationalize any radical denominator, and report the simplified values in the requested order.
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