All 6 Trig Functions

Sine (sin), Cosine (cos), Tangent (tan), Cosecant (csc), Secant (sec), Cotangent (cot).

Sine is the ratio of the length of the opposite side to the length of the hypotenuse: $$\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ .

Cosine is the ratio of the length of the adjacent side to the length of the hypotenuse: $$\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ .

Tangent is the ratio of the length of the opposite side to the length of the adjacent side: $$\text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ .

The relationships can be expressed as: $$\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}, \text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}, \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)}$$ .

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: $$a^2 + b^2 = c^2$$ .

Cotangent is the reciprocal of tangent: $$\text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} = \frac{\text{adjacent}}{\text{opposite}}$$ .

Using the Pythagorean identity: $$\text{cos}^2(\theta) + \text{sin}^2(\theta) = 1$$ , we find $$\text{cos}(\theta) = \frac{4}{5}$$ .

Using the definition of tangent: $$\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} = \frac{3/5}{4/5} = \frac{3}{4}$$ .

Cosecant is the reciprocal of sine: $$\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} = \frac{1}{0.8} = 1.25$$ .