Summary
Highlights
The video introduces the use of double and half-angle formulas for sine, cosine, and tangent. The main challenges are determining the sine and cosine of an unknown angle and identifying the correct quadrant for the half-angle. The process often involves drawing triangles, applying the Pythagorean theorem, and understanding quadrant rules for positive and negative values.
Given cosine theta = 3/5 and theta is in Quadrant 1 (0 to pi/2), the video demonstrates how to find sine and tangent. It then proceeds to calculate sine(2theta), cosine(2theta), tangent(2theta), sine(theta/2), cosine(theta/2), and tangent(theta/2). The key takeaway is that for double angles, the signs are often self-correcting, while for half-angles, explicit quadrant determination is crucial for choosing the positive or negative square root.
This section tackles a more complex scenario where cosecant theta = -sqrt(5) and cosine theta < 0, implying theta is in Quadrant 3. The process involves converting cosecant to sine, using the Pythagorean theorem to find x, and then determining cosine and tangent. The half-angle calculations illustrate how an angle in Quadrant 3, when halved, can fall into Quadrant 2, changing the expected signs of the half-angle trigonometric functions.
The video shows the practical application of half-angle formulas to find exact values for angles not on the unit circle. For example, cosine(22.5 degrees) is re-written as cosine(45/2 degrees), allowing the use of the half-angle formula for cosine with 45 degrees, an angle on the unit circle. Similarly, tangent(9pi/8) is simplified to tangent(pi/8) and then solved as tangent(pi/4 over 2).
Two trigonometric identities are proven. The first, cosine^4(theta) - sine^4(theta) = cosine(2theta), is proven by factoring the left side as a difference of squares and recognizing the Pythagorean identity and the double-angle formula for cosine. The second identity, cosine^2(2theta) - sine^2(2theta) = cosine(4theta), uses the same principle of recognizing the double-angle formula for cosine.
Three equations are solved to demonstrate how to handle double angles. The strategy is to convert the double-angle term into a single angle using the appropriate formula. For cosine(2theta) + 6sine^2(theta) = 4, cosine(2theta) is replaced with 1 - 2sine^2(theta) to match the existing sine^2 term, leading to an equation solvable by isolating sine(theta). For cosine(2theta) = cosine(theta), cosine(2theta) is replaced by 2cosine^2(theta) - 1 to create a quadratic in cosine, which is then factored. Finally, for 2sine(2theta) = cosine(theta), sine(2theta) is replaced by 2sine(theta)cosine(theta), and the equation is solved by factoring out cosine(theta).