Summary
Highlights
The video begins by defining the six trigonometric functions (sine, cosine, tangent, and their reciprocals: cosecant, secant, cotangent) in the context of a right triangle with hypotenuse 'r', adjacent side 'x', and opposite side 'y'. It highlights that for a unit circle, 'r' is 1.
This section reviews the signs of sine, cosine, and tangent in the four quadrants. Sine is positive in quadrants 1 and 2, cosine in 1 and 4, and tangent in 1 and 3. The mnemonic 'All Students Take Calculus' is introduced to remember which functions are positive in each quadrant.
The video demonstrates how to find the six trigonometric functions given a point P(-5, 12) on the terminal side of theta. The point is plotted, a right triangle is formed (identifying a 5-12-13 Pythagorean triple), and each trigonometric function is calculated based on x, y, and r.
Another example is presented with the point P(-8, -15), which falls in the third quadrant. The steps involve plotting the point, recognizing the 8-15-17 Pythagorean triple, and then calculating the six trigonometric functions, confirming that tangent is positive in this quadrant while sine and cosine are negative.
The final example uses the point P(2, -4), which is not a special triangle. The Pythagorean theorem is applied to find 'r' (the hypotenuse), which is 2√5. The video then systematically calculates and rationalizes the six trigonometric functions, paying close attention to simplifying radicals.