Six Trigonometric Ratios

The six trigonometric ratios are: 1. Sine (sin) 2. Cosine (cos) 3. Tangent (tan) 4. Cosecant (csc) 5. Secant (sec) 6. Cotangent (cot)

Sine (sin) of an angle 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 (cos) of an angle 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 (tan) of an angle 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 mnemonic is SOH CAH TOA, which stands for: 1. Sine = Opposite / Hypotenuse (SOH) 2. Cosine = Adjacent / Hypotenuse (CAH) 3. Tangent = Opposite / Adjacent (TOA)

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

Secant (sec) is the reciprocal of cosine: $$\text{sec} \theta = \frac{1}{\text{cos} \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$ .

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

Using the Pythagorean identity: $$\text{cos}^2(y) + \text{sin}^2(y) = 1$$ . Therefore, $$\text{cos}(y) = \frac{18}{18} = \frac{3}{3}$$ .

Using the definition of tangent: $$\text{tan}(y) = \frac{\text{opposite}}{\text{adjacent}}$$ . Therefore, $$\text{sin}(y) = \frac{12}{\text{hypotenuse}}$$ .