Trigonometric Ratios at 0° and 90° MCQs Quiz | Class 10

Understanding Trigonometric Ratios at 0° and 90°

Trigonometry deals with the relationships between the angles and sides of triangles. While we often think of triangles with positive angles, understanding trigonometric ratios at specific angles like 0° and 90° is fundamental. These special angles help define the range and behavior of trigonometric functions and are crucial for solving more complex problems.

Trigonometric Ratios for 0°

Consider a right-angled triangle where one acute angle (let’s say A) approaches 0°. As angle A tends to 0°, the side opposite to angle A (perpendicular) approaches 0, and the adjacent side approaches the hypotenuse in length.

• sin 0°: (Opposite side / Hypotenuse) approaches (0 / Hypotenuse) = 0. So, sin 0° = 0.
• cos 0°: (Adjacent side / Hypotenuse) approaches (Hypotenuse / Hypotenuse) = 1. So, cos 0° = 1.
• tan 0°: (Opposite side / Adjacent side) approaches (0 / Hypotenuse) = 0. So, tan 0° = 0.
• cosec 0°: (Hypotenuse / Opposite side). Since the opposite side approaches 0, cosec 0° = Hypotenuse / 0, which is Undefined.
• sec 0°: (Hypotenuse / Adjacent side) approaches (Hypotenuse / Hypotenuse) = 1. So, sec 0° = 1.
• cot 0°: (Adjacent side / Opposite side). Since the opposite side approaches 0, cot 0° = Hypotenuse / 0, which is Undefined.

Trigonometric Ratios for 90°

Now consider a right-angled triangle where one acute angle (let’s say A) approaches 90°. As angle A tends to 90°, the side adjacent to angle A (base) approaches 0, and the opposite side approaches the hypotenuse in length.

• sin 90°: (Opposite side / Hypotenuse) approaches (Hypotenuse / Hypotenuse) = 1. So, sin 90° = 1.
• cos 90°: (Adjacent side / Hypotenuse) approaches (0 / Hypotenuse) = 0. So, cos 90° = 0.
• tan 90°: (Opposite side / Adjacent side). Since the adjacent side approaches 0, tan 90° = Hypotenuse / 0, which is Undefined.
• cosec 90°: (Hypotenuse / Opposite side) approaches (Hypotenuse / Hypotenuse) = 1. So, cosec 90° = 1.
• sec 90°: (Hypotenuse / Adjacent side). Since the adjacent side approaches 0, sec 90° = Hypotenuse / 0, which is Undefined.
• cot 90°: (Adjacent side / Opposite side) approaches (0 / Hypotenuse) = 0. So, cot 90° = 0.

Summary Table of Trigonometric Ratios for 0° and 90°

Why are some ratios undefined?

A trigonometric ratio is defined as a ratio of sides of a right-angled triangle. When the denominator in such a ratio becomes zero, the ratio is said to be undefined. This occurs when an angle approaches 0° or 90°:

• tan A = sin A / cos A: At A = 90°, cos 90° = 0, so tan 90° is undefined.
• cot A = cos A / sin A: At A = 0°, sin 0° = 0, so cot 0° is undefined.
• cosec A = 1 / sin A: At A = 0°, sin 0° = 0, so cosec 0° is undefined.
• sec A = 1 / cos A: At A = 90°, cos 90° = 0, so sec 90° is undefined.

Understanding these undefined points is crucial for analyzing the domain and range of trigonometric functions.

Quick Revision Points:

• sin 0° = 0, cos 0° = 1, tan 0° = 0
• cosec 0° is Undefined, sec 0° = 1, cot 0° is Undefined
• sin 90° = 1, cos 90° = 0, tan 90° is Undefined
• cosec 90° = 1, sec 90° is Undefined, cot 90° = 0
• Ratios become undefined when division by zero occurs (e.g., opposite side becomes zero for cosec/cot 0°, or adjacent side becomes zero for tan/sec 90°).

Extra Practice Questions:

1. What is the value of (cos 0 degrees – sin 90 degrees)?
2. Find the value of (tan 0 degrees + cot 90 degrees).
3. If theta = 0 degrees, calculate (sin theta + cos theta).
4. Determine if cosec 90 degrees is defined or undefined.
5. Which trigonometric ratio is the reciprocal of sin A and is undefined at 0 degrees?