Pythagorean & Double Angle Identities

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It is expressed as: $$c^2 = a^2 + b^2$$ .

Using the Pythagorean identity $$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$$ , we find that $$\text{cos} \theta = \frac{\text{sqrt}(21)}{5}$$ .

The double angle identities are: 1. $$\text{sin}(2\theta) = 2\text{sin}(\theta)\text{cos}(\theta)$$ 2. $$\text{cos}(2\theta) = \text{cos}^2(\theta) - \text{sin}^2(\theta)$$ .

The solutions are $$\theta = \frac{\text{π}}{4}, \frac{7\text{π}}{4}, \frac{3\text{π}}{4}, \frac{5\text{π}}{4}$$ .

The solutions are $$x = 0, \frac{\text{π}}{3}, \frac{5\text{π}}{3}, 2\text{π}$$ .

Using the double angle identity, $$\text{sin}(2x) = 2\text{sin}(x)\text{cos}(x)$$ . Since $$\text{cos}(x) = \frac{\text{sqrt}(3)}{2}$$ , we find that $$\text{sin}(2x) = \frac{\text{sqrt}(3)}{2}$$ .

In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse.

The range of both the sine and cosine functions is from -1 to 1.

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The coordinates of points on the unit circle correspond to the values of sine and cosine for angles measured from the positive x-axis.

$$\text{cos}(0) = 1$$ and $$\text{sin}(0) = 0$$ .