Geometric Mean, Special Right Triangles, and Trig Ratios(Honors)

The geometric mean is defined as the square root of the product of two numbers: $$\text{Geometric Mean} = \text{sqrt}(a \times b)$$ .

In a 45-45-90 triangle, the lengths of the legs are equal, and the length of the hypotenuse is $$\text{leg} \times \text{sqrt}(2)$$ .

In SOH-CAH-TOA, sine is defined as the ratio of the opposite side to the hypotenuse: $$\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ .

In SOH-CAH-TOA, cosine is defined as the ratio of the adjacent side to the hypotenuse: $$\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ .

In SOH-CAH-TOA, tangent is defined as the ratio of the opposite side to the adjacent side: $$\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ .

To find the length of the hypotenuse, use the Pythagorean theorem: $$c = \text{sqrt}(a^2 + b^2)$$ , where c is the hypotenuse and a and b are the legs.

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1 : $$\text{sqrt}(3) : 2$$ .

The sine of a 30-degree angle is $$\frac{1}{2}$$ .

The cosine of a 60-degree angle is $$\frac{1}{2}$$ .

The tangent of a 45-degree angle is $$1$$ .