Existence/Well-defined Ratios MCQs Quiz | Class 10

Understanding Well-Defined Trigonometric Ratios

Trigonometric ratios are fundamental concepts in mathematics, especially in the study of triangles and angles. The phrase “well-defined” in this context is crucial, as it ensures that these ratios have consistent and predictable values, making trigonometry a powerful tool for measurement and analysis.

What Does “Well-Defined” Mean?

When we say a trigonometric ratio (like sine, cosine, or tangent) is well-defined for a fixed angle, it means that its value is constant regardless of the size of the right-angled triangle from which it is calculated. If you have a 30-degree angle, its sine value will always be 0.5, whether it’s in a small triangle or a very large one. This consistency allows us to create tables and use calculators for these values.

Why are Ratios Well-Defined for a Fixed Angle? The Role of Similar Triangles

The core reason behind the well-defined nature of trigonometric ratios lies in the concept of similar triangles. Here’s a breakdown:

1. Similarity of Right-Angled Triangles: Consider any two right-angled triangles that both share the same acute angle (let’s call it angle theta). Since both triangles have a 90-degree angle and a common acute angle, their third angles must also be equal (because the sum of angles in a triangle is 180 degrees). This satisfies the Angle-Angle (AA) similarity criterion. Therefore, all right-angled triangles with the same acute angle are similar to each other.
2. Constant Ratios of Corresponding Sides: A key property of similar triangles is that the ratios of their corresponding sides are equal. For example, if triangle ABC is similar to triangle PQR, then AB/PQ = BC/QR = AC/PR. When applied to trigonometric ratios, this means:
   • Sine (Opposite/Hypotenuse): If we consider the ratio of the side opposite to angle theta to the hypotenuse in both similar triangles, these ratios will be equal.
   • Cosine (Adjacent/Hypotenuse): Similarly, the ratio of the adjacent side to the hypotenuse will be the same.
   • Tangent (Opposite/Adjacent): The ratio of the opposite side to the adjacent side will also be identical.
3. Independence from Triangle Size: Because these ratios depend only on the angles (specifically, the fixed acute angle) and not on the absolute lengths of the sides, the size of the right-angled triangle does not affect the value of the trigonometric ratio. Scaling a triangle up or down proportionally changes all its side lengths, but the ratios between those corresponding sides remain constant.

Quick Revision Points

• Trigonometric ratios are constant for a specific acute angle.
• This constancy is due to the property of similar triangles.
• All right triangles with the same acute angle are similar.
• Ratios of corresponding sides in similar triangles are equal.
• The size of the triangle does not influence the ratio for a given angle.

Extra Practice Questions

Test your understanding with these additional questions:

1. Explain in your own words why sine of 60 degrees is always the same value, regardless of the right triangle used.
2. If two right triangles have one acute angle equal to 50 degrees, what can you conclude about their relationship?
3. A student measures the adjacent side and hypotenuse of a small right triangle to find the cosine of an angle. If they use a larger right triangle with the same angle, will the cosine value change? Justify your answer.
4. What happens to the trigonometric ratios if only one side of a right triangle is scaled, while the angles are kept fixed?
5. Provide an example where the well-defined nature of trigonometric ratios is useful in real-world applications.