Mathematics — AI-generated practice question

Trigonometric ratios are defined as ratios of sides of a right triangle with respect to an angle. For example, $\sin A = \dfrac{\text{opposite side}}{\text{hypotenuse}}$.

Using Similar Triangles:

Consider right triangle ABC with acute angle A. Take a point P on hypotenuse AC and draw PM ⊥ AB, forming a smaller right triangle PAM. Both triangles PAM and CAB share angle A and have a right angle each. By the AA similarity criterion, △PAM ~ △CAB.

By the property of similar triangles, corresponding sides are proportional:

$$\frac{AM}{AB} = \frac{AP}{AC} = \frac{MP}{BC}$$

From this:

$$\frac{MP}{AP} = \frac{BC}{AC} = \sin A \qquad \text{and} \qquad \frac{AM}{AP} = \frac{AB}{AC} = \cos A$$

This shows that $\sin A$ and $\cos A$ in △PAM equal those in △CAB, even though the side lengths differ.

Since the ratios depend only on the angle A (not on the actual lengths), all triangles with the same acute angle A are similar to each other, and their corresponding side ratios remain constant. Hence, the values of trigonometric ratios of an angle do not vary with the lengths of the sides, as long as the angle remains the same.

Source: Chapter 8, Section 8.2

---

• The examiner expects you to: (1) state that trig ratios are ratios of sides, (2) invoke AA similarity criterion explicitly, (3) write the proportionality condition, and (4) extract at least two trig ratios to show they remain unchanged.
• The key logical chain is: same angle → AA similarity → proportional sides → same ratios. Don't just state the conclusion; show the steps.
• Mentioning that any point on the hypotenuse gives a similar triangle (△QAN also works) earns full marks.
• Avoid writing the full proof for all six ratios; two or three examples suffice for 5 marks.