What is the Value of Sin 45 in Trigonometry?
The concept of sin 45 value is an essential part of trigonometry in mathematics, helping students solve right triangle problems and making exam calculations faster and more accurate.
What Is Sin 45 Value?
The sin 45 value refers to the sine of a 45-degree angle, which is a standard trigonometric ratio important for solving various mathematical and real-life problems. Sine values are commonly used in calculating sides and angles of triangles, physics equations, and engineering designs. In school mathematics, knowing sin and cos values at special angles like 45° is especially useful for board and entrance exams.
Key Formula for Sin 45 Value
Here’s the standard formula used to calculate the value of sin 45:
\( \sin 45^\circ = \frac{1}{\sqrt{2}} \) or \( \sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071 \)
| Angle | sin (θ) | cos (θ) | tan (θ) |
|---|---|---|---|
| 45° or π/4 |
1/√2 or √2/2 0.7071 |
1/√2 or √2/2 0.7071 |
1 |
Cross-Disciplinary Usage
The sin 45 value is not only significant in Mathematics, but also appears in Physics, Computer Science, Geography, and Engineering. For instance, it is used in vector resolution, calculating heights and distances, wave equations, signal processing, and even computer graphics for rotations. Students preparing for exams like JEE, NEET, or Olympiads will find this value repeatedly in MCQ and problem-solving sections.
Step-by-Step Illustration
-
Consider an isosceles right triangle with both legs equal to 1 unit.
By the Pythagorean theorem:
Hypotenuse = \( \sqrt{1^2 + 1^2} = \sqrt{2} \) -
The sine of 45° is the ratio of the opposite side to the hypotenuse:
\( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
-
To rationalize the denominator, multiply numerator and denominator by \( \sqrt{2} \):
\( \sin 45^\circ = \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{\sqrt{2}}{2} \)
- In decimal form, \( \sin 45^\circ = 0.7071 \) (rounded to four decimal places).
Speed Trick or Vedic Shortcut
Here's a memory trick for remembering sin 45 value quickly: All sine values for standard angles (0°, 30°, 45°, 60°, 90°) can be found using:
Formula: \( \sin \theta = \sqrt{n}/2 \) where n = 0, 1, 2, 3, 4 for angles 0°, 30°, 45°, 60°, 90°.
- For 45°, n = 2, so \( \sqrt{2}/2 \).
- Repeat for cos values, but in reverse angle order.
Tricks like these are highly useful for MCQ exams and fast mental calculations. Teachers at Vedantu often recommend making a quick revision table before your test!
Try These Yourself
- Write the value of sin 45° in fraction, decimal, and root forms.
- Solve: If the hypotenuse of a right triangle is 10 cm and the angle is 45°, what is the length of the side opposite to 45°?
- Calculate the value of sin 45° + cos 45°.
- Is tan 45° equal to 1? Why?
Frequent Errors and Misunderstandings
- Confusing sin 45 value with sin 30 or sin 60 values.
- Forgetting to rationalize \( \frac{1}{\sqrt{2}} \) as \( \frac{\sqrt{2}}{2} \).
- Mixing degree and radian measures (45° and π/4 radians are the same angle).
- Copying decimal values incorrectly—always round 0.7071 to four digits in calculations.
Relation to Other Concepts
The sin 45 value connects to many other trigonometric topics, such as the trigonometric identities, standard trigonometry tables, and the unit circle. Understanding sin 45 also helps in grasping complementary angles because sin 45° = cos 45°.
Classroom Tip
A helpful way to remember sin 45 value is to recall that in a square cut in half diagonally, the sides and the diagonal form a 45-45-90 triangle. Both the perpendicular and the base are equal, so sin and cos values are the same for 45°. Vedantu’s live online sessions make it even easier by providing visual diagrams and memory aids.
Wrapping It All Up
We explored sin 45 value—including its exact value, formula, stepwise proofs, memory tips, and why it's relevant in maths and physics. Practicing problems using this value makes you faster and more confident during exams. For more on trigonometric values and practice problems, check out Vedantu’s resources and interactive classes.
Related Topics and Further Practice
- Cos 45 Value: See why cos 45° is equal to sin 45° and how to use both in proofs.
- Sin 30 Degrees: Compare sin 45° to sin 30° for better memorization.
- Trigonometric Table: Find all your standard angle values in one place.
- Unit Circle: Learn about radians (π/4) and see how sin 45° fits on the graph.
- Trigonometric Identities: Practice how sin 45° is used in different identities.
FAQs on Sin 45 Value Explained for Students
1. What is the value of sin 45 degrees?
The value of sin 45 degrees is 1/√2, which is approximately equal to 0.7071. This value is fundamental in trigonometry and is frequently used in calculations involving right-angled triangles.
2. How do I calculate sin 45 degrees without a calculator?
You can calculate sin 45 degrees using the properties of a 45-45-90 isosceles right-angled triangle. In such a triangle, the two legs are equal in length, and the hypotenuse is √2 times the length of a leg. Therefore, the sine of 45 degrees (opposite/hypotenuse) simplifies to 1/√2. You can then rationalize this to get √2/2.
3. What is sin 45 degrees in radians?
45 degrees is equivalent to π/4 radians. Therefore, sin(45°) = sin(π/4) = 1/√2.
4. How is sin 45 degrees used in geometry problems?
Sin 45 degrees is crucial for solving problems involving isosceles right-angled triangles. It allows you to find the lengths of sides or angles when you know other side lengths or angles. It is also used in more complex geometric problems that incorporate trigonometry.
5. What are some common mistakes students make when working with sin 45 degrees?
Common mistakes include: forgetting to rationalize the denominator of 1/√2; confusing the value of sin 45 degrees with other trigonometric ratios like sin 30 degrees or sin 60 degrees; and misremembering the decimal approximation of 1/√2.
6. Why is sin 45° equal to cos 45°?
In a 45-45-90 triangle, the opposite and adjacent sides to the 45° angle are equal in length. Since sine is opposite/hypotenuse and cosine is adjacent/hypotenuse, and the opposite and adjacent sides are equal, then sin 45° and cos 45° are also equal, both equaling 1/√2.
7. How can I remember the value of sin 45 degrees easily?
Visualize an isosceles right triangle with legs of length 1. The hypotenuse will be √2, and the sine will be 1/√2. Alternatively, memorize the decimal approximation: 0.7071. Regular practice with problems involving sin 45° will also solidify its value in your memory.
8. What is the relationship between sin 45 degrees and the Pythagorean identity?
The Pythagorean identity (sin²θ + cos²θ = 1) holds true for all angles, including 45 degrees. Since sin 45° = cos 45° = 1/√2, substituting these values into the identity demonstrates its validity: (1/√2)² + (1/√2)² = 1.
9. How does sin 45 degrees relate to the unit circle?
On the unit circle, a 45° angle intersects the circle at the point (√2/2, √2/2). The y-coordinate of this point represents sin 45°, which is √2/2 or 1/√2.
10. Can sin 45 degrees be expressed exactly without using a decimal approximation?
Yes, the exact value of sin 45 degrees is 1/√2 or its rationalized equivalent, √2/2. These expressions are precise and do not involve rounding errors inherent in decimal approximations.
11. How can I use sin 45° to solve real-world problems?
Sin 45° is useful in various applications including calculating heights and distances in surveying, finding angles in architecture and engineering, and determining projectile motion trajectories in physics. Many scenarios involving right-angled triangles rely on this fundamental trigonometric ratio.
