Instructions.
In a right-angled triangle, the two acute angles always sum to \(90^\circ\text{.}\) Use the slider to change the angle \(\theta\) and observe the results:
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Observe the Angles: As you increase \(\theta\) (the blue angle), what happens to the size of its complement, \(\phi\) (the green angle)? What is their sum at any given point?
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Track the Sides: Look at the side \(BC\text{.}\) Relative to the blue angle (\(\theta\)), is this the "Opposite" or "Adjacent" side? Now, look at it relative to the green angle (\(\phi\)). What role does it play there?
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Compare the Ratios: Compare the value of \(\sin(\theta)\) in the left column to \(\cos(\phi)\) in the right column. Why do you think these two different trigonometric functions result in the exact same decimal value?
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Find the Crossing Point: Move the slider until \(\theta\) is exactly \(45^\circ\text{.}\) What do you notice about all four ratios (Sine and Cosine for both angles) at this specific position?