Why is Sin 90 Degrees Equal to 1? (Proof & Concept)
The concept of sin 90 degrees plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are working with triangles, the unit circle, or preparing for competitive exams like JEE or NEET, knowing the value and understanding the formula of sin 90° makes solving trigonometry problems quick and accurate.
What Is Sin 90 Degrees?
Sin 90 degrees is the value of the sine trigonometric function when the angle is exactly 90 degrees. In simple terms, the sine of 90 degrees answers the question: what is the ratio of the length of the side opposite to a 90° angle, to the triangle’s hypotenuse? You’ll find this concept often in geometry, trigonometric ratios, and when analyzing the unit circle.
Key Formula for Sin 90 Degrees
Here’s the standard formula: \( \sin(90^\circ) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)
Since the opposite side equals the hypotenuse in a right-angled triangle when angle A = 90°, the answer becomes:
\( \sin(90^\circ) = \frac{1}{1} = 1 \)
Sin 90 Degrees on the Unit Circle
On the unit circle, sin 90° represents the y-coordinate of the point where the terminal side of a 90° angle meets the unit circle. On this circle, with radius 1, the (x, y) coordinate at 90° is (0, 1), so the value is 1.
Sin 90 Degrees in Radians & Fractions
| Angle | Degrees | Radians | Fraction | Sin Value |
|---|---|---|---|---|
| 90° | 90 | π/2 | 1/1 | 1 |
So, sin 90 degrees is the same as sin π/2 radians, and in fraction form, it’s just 1.
Sin 90 Degrees Value Table (0° to 360°)
| Angle | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
This table is useful for quick reference, especially in MCQs. Remember, sin 90 degrees is always 1.
Why Is Sin 90 Degrees Equal to 1? (Step-by-Step Proof)
Here’s how you can prove that sin 90° = 1 using the unit circle method:
1. Draw a unit circle with center at the origin and radius = 1.2. Mark the positive x-axis as 0°, and the positive y-axis as 90°.
3. The coordinates for 90° on this circle are (0, 1).
4. Sine of any angle equals the y-coordinate, so sin 90° = 1.
You can also see this in a right triangle where the angle is 90°. The side opposite the angle is as long as the hypotenuse, so their ratio is 1.
Cross-Disciplinary Usage
Sin 90 degrees is not only useful in Maths but also plays an important role in Physics, Engineering, and Computer Science. For instance, in wave motion and circular motion, sin 90° often represents the maximum displacement. Students preparing for JEE or NEET will see its relevance in geometry, trigonometric identities, and scientific problems where a value reaches its maximum.
Examples Using Sin 90 Degrees
Example 1: Solve: (1 − cos²(90°))
1. cos 90° = 02. (1 − (0)²) = 1 − 0 = 1
Example 2: Simplify: 2 × (sin 90°/sin 270°)
1. sin 90° = 1, sin 270° = −12. 2 × (1/−1) = 2 × (−1) = −2
Example 3: In a right-angled triangle, what is the value of sin 90°?
1. Angle = 90°, so sin 90° = 1 (by definition).Try These Yourself
- Find the value of sin 0°, sin 30°, sin 45°, sin 60°, and sin 90°.
- If the hypotenuse of a right triangle is 6 cm, and the angle is 90°, what is the length of the opposite side?
- Simplify: sin 90° + cos 90°
- What is the value of sin (π/2) in radians?
Frequent Errors and Misunderstandings
- Students sometimes confuse sin 90° with cos 90°, but remember: sin 90° = 1, cos 90° = 0.
- Using the wrong mode on calculators (degrees vs radians) can give incorrect answers.
- Memorizing values without understanding the unit circle proof.
Relation to Other Concepts
The idea of sin 90 degrees connects closely with topics such as cos 90 degrees, sin 0 degrees, and the trigonometric ratios of standard angles. Mastering these values helps with identities, equations, and advanced coordinate geometry.
Classroom Tip
A quick way to remember sin 90° is "sine starts at zero (0°), reaches the top at 90° (1), and repeats every 360°." Vedantu’s teachers often use the 'finger trick' or the unit circle diagram in class to help children visualize and recall these values easily.
We explored sin 90 degrees—from definition, formula, value on the unit circle, example problems, and tips to avoid mistakes. Continue practicing these concepts with Vedantu to become confident and fast in trigonometry for any exam or classroom activity!
Cos 90 Degrees | Unit Circle | Trigonometric Ratios Table | Sin 0 Degrees
FAQs on Sin 90 Degrees: Value, Formula & Proof Explained
1. What is the value of sin 90 degrees?
The value of sin 90 degrees is 1. This is a fundamental trigonometric value often used in calculations and problem-solving.
2. How do you prove that sin 90° = 1?
The proof relies on the definition of sine in a right-angled triangle or on the unit circle. In a right-angled triangle, sin θ = opposite/hypotenuse. As the angle approaches 90°, the opposite side becomes equal to the hypotenuse, resulting in a ratio of 1. On the unit circle, the y-coordinate at 90° is 1, which represents the sine value.
3. What is sin 90 degrees in radians?
90 degrees is equivalent to π/2 radians. Therefore, sin(π/2) = 1.
4. What are some common trigonometric identities involving sin 90°?
Several identities utilize sin 90°:
• sin(90° + θ) = cos θ
• sin(90° - θ) = cos θ
• sin²θ + cos²θ = 1 (This fundamental identity is particularly useful when combined with the value of sin 90°).
5. How is the sin 90° value used in solving trigonometric problems?
The value of sin 90° simplifies equations and allows for direct substitutions, especially in problems involving right-angled triangles or unit circle applications. Knowing this value significantly reduces calculation steps.
6. What is the difference between sin 90° and cos 90°?
Sin 90° = 1 and cos 90° = 0. These values represent the y and x coordinates, respectively, of the point on the unit circle at 90°.
7. What is the maximum value of the sine function, and where does it occur?
The maximum value of the sine function is 1, and it occurs at 90° (or π/2 radians) and its coterminal angles.
8. Can sin 90° ever be negative?
No, in standard trigonometry, sin 90° is always positive (1). A negative result might indicate an error in angle measurement (e.g., incorrect use of radians/degrees) or calculator settings.
9. How do I use the sin 90° value in geometry problems?
In geometry problems involving right-angled triangles, sin 90° can simplify calculations, particularly when using trigonometric ratios (SOH CAH TOA) to find missing sides or angles.
10. What is the relationship between sin 90° and the unit circle?
The unit circle provides a visual representation. The sine of an angle is its y-coordinate on the unit circle. At 90°, this y-coordinate is 1, thus sin 90° = 1.
11. Are there any practical applications of sin 90° in real-world scenarios?
Yes. Many fields use trigonometry, such as surveying, navigation, and physics (projectile motion). Understanding sin 90° is crucial for solving problems in these areas.
12. How does sin 90° relate to other trigonometric functions at 90°?
At 90°, sin 90° = 1, cos 90° = 0, and tan 90° is undefined. These values are interconnected through trigonometric identities.
