SOHCAHTOA

• The principle of similarity dictates that for any right-angled triangle with a specific angle $\theta$, the ratio of its side lengths remains constant regardless of the triangle's overall size.

• The Sine ratio ($S = O/H$) describes the vertical proportion of the triangle relative to the hypotenuse.

• The Cosine ratio ($C = A/H$) describes the horizontal proportion of the triangle relative to the hypotenuse.

• The Tangent ratio ($T = O/A$) describes the slope or gradient of the hypotenuse relative to the base.

Finding a Missing Side Length

• Step 1: Label the sides (H, O, A) based on the given angle and the side you need to find.

• Step 2: Select the correct ratio (SOH, CAH, or TOA) that includes both the known side and the unknown side.

• Step 3: Substitute the known values into the formula and rearrange to isolate the unknown variable.

Finding a Missing Angle

• Step 1: Label the two known sides as H, O, or A.

• Step 2: Identify which ratio uses these two sides.

• Step 3: Use the inverse trigonometric function (e.g., $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) on your calculator to solve for the angle.

• Opposite vs. Adjacent: The 'Opposite' side is always across from the angle, while the 'Adjacent' side is the leg forming the angle. If you switch the reference angle, these two labels swap places.

• Non-Right Triangles: A frequent error is attempting to use SOHCAHTOA on triangles that do not have a 90-degree angle. In such cases, the Sine Rule or Cosine Rule must be used instead.

• Incorrect Side Labeling: Students often mislabel the Adjacent side as the Hypotenuse because both sides touch the angle $\theta$. Always identify the Hypotenuse first as the side opposite the right angle.

• Algebraic Errors: When the unknown is in the denominator (e.g., $\cos(30) = \frac{5}{x}$), students often multiply incorrectly. The correct rearrangement is $x = \frac{5}{\cos(30)}$.