Right-Angled Triangles

• Sine Ratio (SOH): Defined as the ratio of the opposite side to the hypotenuse, expressed as $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$.

• Cosine Ratio (CAH): Defined as the ratio of the adjacent side to the hypotenuse, expressed as $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.

• Tangent Ratio (TOA): Defined as the ratio of the opposite side to the adjacent side, expressed as $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$.

• Geometric Consistency: These ratios remain constant for a specific angle regardless of the triangle's size, which is why they are fundamental functions in mathematics.

Step-by-Step Problem Solving

• Step 1: Label the sides: Identify the Hypotenuse, Opposite, and Adjacent sides relative to the angle provided or required.
• Step 2: Select the ratio: Choose between Sine, Cosine, or Tangent based on which two sides are involved in the problem (one known and one unknown).
• Step 3: Substitute and Solve: Plug the known values into the chosen formula and rearrange the equation to isolate the unknown variable.
• Step 4: Inverse Functions: When solving for an angle, use the inverse trigonometric functions ($\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) on your calculator to find the degree measure.

• Static Labeling: Students often mistakenly think the 'Opposite' side is always the vertical one; it actually changes depending on which of the two non-right angles is being used as the reference.

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

• Misusing SOHCAHTOA: These specific ratios only apply to triangles with a $90^{\circ}$ angle; applying them to scalene or isosceles triangles without a right angle will result in incorrect answers.