Trigonometric Ratios at 0° and 90° MCQs Quiz | Class 10
This quiz is designed for Class X, Subject: Mathematics (Code 041), part of Unit V: Trigonometry, covering the Topic: Trigonometric Ratios at 0° and 90°. It focuses on understanding which ratios are defined or undefined at these angles and the reasoning behind them. Attempt all 10 multiple-choice questions, then submit to see your results and download a detailed answer PDF for revision.
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°
| Angle | sin A | cos A | tan A | cosec A | sec A | cot A |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Undefined | 1 | Undefined |
| 90° | 1 | 0 | Undefined | 1 | Undefined | 0 |
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:
- What is the value of (cos 0 degrees – sin 90 degrees)?
- Find the value of (tan 0 degrees + cot 90 degrees).
- If theta = 0 degrees, calculate (sin theta + cos theta).
- Determine if cosec 90 degrees is defined or undefined.
- Which trigonometric ratio is the reciprocal of sin A and is undefined at 0 degrees?
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