1. We will list the main geometric formulas involving the trigonometric functions sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). 2. **Basic definitions in a right triangle:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ - $\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan \theta}$ - $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$ - $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$ 3. **Pythagorean identities:** - $\sin^2 \theta + \cos^2 \theta = 1$ - $1 + \tan^2 \theta = \sec^2 \theta$ - $1 + \cot^2 \theta = \csc^2 \theta$ 4. **Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 5. **Law of Cosines:** $$c^2 = a^2 + b^2 - 2ab \cos C$$ (similarly for other sides) 6. **Area of a triangle using sine:** $$\text{Area} = \frac{1}{2}ab \sin C$$ 7. **Sum and difference formulas:** - $\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$ - $\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$ - $\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ These formulas are fundamental in geometry and trigonometry for solving triangles and analyzing angles and sides.