Advanced Trigonometric Identities: Double Angle, Half Angle, Product-to-Sum

Trigonometric Identities and Applications

Review of Trigonometric Basics & Algebraic Manipulation

• Finding Trigonometric Values from a Triangle:

  • Given a scenario where \sin is -3/5, implying the y-coordinate is -3 and the hypotenuse is -5 (or just -3 and -5 for y and r in a coordinate plane), the other side (x-coordinate) can be found using the Pythagorean theorem, x^2 + (-3)^2 = 5^2, leading to x^2 + 9 = 25, x^2 = 16, and x = \pm 4. The sign depends on the quadrant.

  • In another example, if x = -7 and the hypotenuse r = 25, then (-7)^2 + y^2 = 25^2. So, 49 + y^2 = 625, y^2 = 576, and y = \pm \sqrt{576} = \pm 24. If A is the angle, then \sin A = y/r = 24/25 (adjusting for quadrant if necessary, as speaker mentione