Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

What is the Value of Sin 180° in Maths?

Reviewed by:
ffImage
hightlight icon
highlight icon
highlight icon
share icon
copy icon
SearchIcon

Why is Sin 180 Degrees Equal to Zero? Proof & Explanation

The concept of value of sin 180 is fundamental in mathematics, especially within trigonometry. Knowing the sin 180 value allows students to quickly answer MCQs, analyze graphs, and solve real-world problems related to waves and periodic motion. It also forms the basis for many formulas in exams and competitions.


What Is Value of Sin 180?

The value of sin 180 is zero. In trigonometry, “sin” stands for the sine function, and 180 refers to 180 degrees (or π radians). On the unit circle, the sine of any angle equals the y-coordinate of the corresponding point. At 180°, this point is at (-1, 0), making the y-coordinate (and thus sin 180 degrees) equal to 0. You’ll find this concept applied throughout geometry, physics, and competitive exam questions.


Key Formula for Value of Sin 180

Here’s the standard formula: \( \sin 180^\circ = 0 \)
This result holds true whether the angle is in degrees (180°) or radians (π radians): \( \sin \pi = 0 \).


Value Table for Sin, Cos, Tan at Key Angles

Angle (°) Angle (rad) sin cos tan
0 0 0 1 0
30 π/6 1/2 √3/2 1/√3
90 π/2 1 0 Undefined
180 π 0 -1 0
360 0 1 0

Step-by-Step Illustration

Let’s prove the value of sin 180:

1. Recall that on the unit circle, any angle θ has coordinates (cos θ, sin θ).

2. At 180°, the point is at (-1, 0).

3. So, sin 180° is the y-coordinate = 0.

4. So, sin 180° = 0.

Alternate proof using formula:
sin 180° = sin(90° + 90°) = cos 90° = 0.


Speed Trick or Vedic Shortcut

For rapid recall in exams, remember the patterns for sine at standard angles:

  • sin 0° = 0
  • sin 90° = 1
  • sin 180° = 0
  • sin 270° = -1
  • sin 360° = 0

Just like a wave touching the baseline at every multiple of 180°. When unsure, always check the unit circle, or draw the graph to see where sine hits zero! Vedantu sessions often use such patterns to train for speed-based MCQ rounds.


Cross-Disciplinary Usage

The value of sin 180 is important beyond mathematics. In physics, it explains wave symmetry, alternating current cycles, and oscillations. In computer science, understanding periodic signals involves these trigonometric values. Students facing exams like JEE, NEET, or board tests routinely use sin 180 value in calculations and proofs.


Relation to Other Trigonometric Values

Function Value at 180° Explanation
sin 180° 0 Y–coordinate on unit circle
cos 180° -1 X–coordinate on unit circle
tan 180° 0 sin/cos = 0/−1 = 0

Understanding these together helps avoid confusion in exams and pushes accuracy in problem solving.


Solved Examples Using Sin 180 Value

Example 1: Find the value of 2 × (sin 90°) × (cos 90°) + sin 180°.
Solution:
2 × (1) × (0) + 0 = 0 + 0 = 0

Example 2: Simplify: sin(180°–x) + sin(180°+x).
Solution:
sin(180°–x) = sin x
sin(180°+x) = –sin x
Sum = sin x + (–sin x) = 0

Example 3: What is the value of sin(π)?
Solution:
π radians = 180°.
Therefore, sin(π) = sin(180°) = 0.


Try These Yourself

  • What is sin 180° in radians?
  • Fill in the blank: sin 0° = __, sin 90° = __, sin 180° = __
  • Prove that sin(180°–x) = sin x
  • Is sin 180° positive, negative, or zero?
  • Where does the graph of sin x cut the x-axis between 0° and 360°?

Frequent Errors and Misunderstandings

  • Mistaking sin 180° for 1 or -1 (actually 0!)
  • Mixing up degree and radian measure: sin(π) = sin(180°), both are 0
  • Forgetting that sin (n × 180°) is always zero for integer n
  • Thinking sin 180 is negative—in fact, at 180°, the sine switches to 0

Classroom Tip

A quick way to remember the value of sin 180 is to visualize the unit circle: at 180°, the point lands on the negative x-axis (leftmost point), where the height (y) is zero. Vedantu’s teachers use this circle method during live classes for strong concept recall.


Relation to Other Concepts

Mastering the value of sin 180 helps in trigonometric proofs, identities, and advanced applications in calculus and vector analysis. It also harmonizes with Trigonometric Ratios of Standard Angles and Trigonometric Functions needed for higher studies.


We explored value of sin 180—from direct answer, formula, stepwise proof, examples, mistakes, and links to other key topics. Consistent practice with Vedantu’s resources will make you confident with all trigonometric values and their usage in any maths exam.


Best Seller - Grade 12 - JEE
View More>
Previous
Next

FAQs on What is the Value of Sin 180° in Maths?

1. What is the exact value of sin 180° in trigonometry?

The exact value of sin 180° is 0. This is a fundamental value in trigonometry, representing the sine function at a straight angle. It is crucial for solving various mathematical problems and understanding the behaviour of trigonometric functions.

2. Why is the value of sin 180° equal to zero?

The value of sin 180° is zero because of its definition on the unit circle. The sine of an angle corresponds to the y-coordinate of the point where the angle's terminal side intersects the circle. At 180°, this point is (-1, 0) on the negative x-axis, where the y-coordinate is precisely 0.

3. How can the value of sin 180° be explained using the unit circle?

Using the unit circle (a circle with a radius of 1):

  • Start from the positive x-axis (0°).

  • Rotate counter-clockwise by 180°.

  • The terminal side of the angle lies on the negative x-axis.

  • The coordinates of this intersection point are (-1, 0).

  • Since the sine function represents the y-coordinate, sin 180° is equal to 0.

4. What is the value of sin 180° in radians?

An angle of 180° is equivalent to π radians. Therefore, the value of sin 180° is the same as sin(π), which is 0. The value of the trigonometric function remains the same regardless of whether the angle is measured in degrees or radians.

5. How do the values of sin 180° and cos 180° differ?

The key difference lies in which coordinate they represent on the unit circle. While sin 180° = 0 (representing the y-coordinate at the point (-1, 0)), cos 180° = -1 (representing the x-coordinate at the same point). They describe different dimensional components of the same angular position.

6. What is the significance of sin 180° = 0 in the sine wave graph?

In the graph of the function y = sin(x), the value of sin 180° (or π radians) represents a zero-crossing point. This is where the wave intersects the horizontal x-axis, completing its first positive arch and moving into the negative region. It signifies a point of equilibrium or zero displacement in wave phenomena.

7. How is the value of sin 180° used to simplify the identity for sin(180° - θ)?

The angle subtraction formula for sine is sin(A - B) = sin(A)cos(B) - cos(A)sin(B). By substituting A = 180°, we get:
sin(180° - θ) = sin(180°)cos(θ) - cos(180°)sin(θ)
Since sin(180°) = 0 and cos(180°) = -1, the equation simplifies to:
sin(180° - θ) = (0)cos(θ) - (-1)sin(θ) = sin(θ).

8. What are some real-world applications where the concept of sin 180° is important?

The concept is fundamental in many fields. For example:

  • Physics: In wave mechanics, a phase shift of 180° (or π) results in destructive interference, where two waves cancel each other out.

  • Engineering: In AC circuits, it helps describe points where the voltage or current is zero.

  • Simple Harmonic Motion: It defines the point where an oscillating object passes through its equilibrium position while changing direction.

9. Are the values of sin 180° and sin 360° the same?

Yes, both sin 180° and sin 360° are equal to 0. On the unit circle, 180° corresponds to the point (-1, 0) and 360° corresponds to the point (1, 0). In both cases, the y-coordinate is zero. These represent the two points where the sine function's graph crosses the x-axis in one full cycle.

10. What is a common misconception students have when comparing sin 180° and sin 90°?

A common misconception is thinking that a larger angle implies a larger sine value. However, sin 90° = 1, which is the maximum value of the sine function, representing the peak of the sine wave. In contrast, sin 180° = 0, representing the point where the wave returns to the horizontal axis. The value of sine depends on the position in the cycle, not just the magnitude of the angle.