Sin 60 Value: Complete Explanation & Examples for Students

What is Sin 60°?

The sin 60 value is one of the fundamental values in trigonometry. It represents the sine of an angle measuring 60 degrees.

The exact value of sin 60 is:

sin 60° = √3/2 = 0.866025...

This means that if we take an angle of 60 degrees in a right-angled triangle, the ratio of the opposite side to the hypotenuse equals √3/2 (approximately 0.866).

Why is Sin 60 Equal to √3/2?

To understand why sin 60 equals √3/2, we can use several approaches:

1. Using the Unit Circle

In the unit circle (a circle with radius 1), the sin 60 value represents the y-coordinate of the point where the terminal arm of a 60° angle intersects the circle.

When we place a 60° angle in standard position on the unit circle:

• The x-coordinate is 1/2 (which is cos 60°)
• The y-coordinate is √3/2 (which is sin 60°)

2. Using an Equilateral Triangle

We can find the sin 60 value using an equilateral triangle with sides of length 2:

1. An equilateral triangle has three 60° angles
2. Draw a height from one vertex to the opposite side
3. This creates two right triangles
4. In each right triangle:
   • The hypotenuse is 2
   • One angle is 60°
   • The side opposite to the 60° angle is the height
5. Using the Pythagorean theorem, the height equals √3
6. Therefore, sin 60 = opposite/hypotenuse = √3/2

Exact and Decimal Value of Sin 60

It's important to know both forms of the sin 60 value:

• Exact value: sin 60° = √3/2
• Decimal approximation: sin 60° ≈ 0.866025...

In mathematics, we often prefer to use the exact value (√3/2) rather than the decimal approximation because it's precise and maintains the relationship with other trigonometric values.

Sin 60 in Radians

The angle 60° can also be expressed in radians:

60° = π/3 radians

So, sin 60° is equivalent to sin(π/3) = √3/2

Examples Using Sin 60

Let's explore various examples that use the sin 60 value:

Example 1: Finding the Height of a Right Triangle

Problem: In a right triangle, one angle is 60° and the hypotenuse is 10 cm. Find the side opposite to the 60° angle.

Solution:

1. sin 60° = opposite/hypotenuse
2. opposite = hypotenuse × sin 60°
3. opposite = 10 × (√3/2)
4. opposite = 5√3 cm ≈ 8.66 cm

Example 2: Finding a Length in an Equilateral Triangle

Problem: An equilateral triangle has sides of length 8 cm. Find its height.

Solution:

1. In an equilateral triangle, the height divides it into two right triangles
2. In each right triangle, the angle at the base is 60°
3. The height is opposite to this 60° angle
4. sin 60° = height/(side length)
5. height = (side length) × sin 60°
6. height = 8 × (√3/2)
7. height = 4√3 cm ≈ 6.93 cm

Example 3: Finding the Area Using Sin 60

Problem: Find the area of a rhombus with diagonals of lengths 10 cm and 20 cm.

Solution:

1. The diagonals of a rhombus bisect each other at right angles
2. This creates four right triangles
3. In each right triangle, the angle between the diagonals is 60°
4. Area of rhombus = (d? × d?)/2
5. Area = (10 × 20)/2 = 100 cm²

Example 4: Simple Harmonic Motion

Problem: A pendulum oscillates with amplitude 5 cm and angular frequency 2 rad/s. Find its displacement when t = π/6 seconds.

Solution:

1. Displacement = A sin(ωt)
2. Displacement = 5 sin(2 × π/6)
3. Displacement = 5 sin(π/3)
4. Since sin 60° = sin(π/3) = √3/2
5. Displacement = 5 × (√3/2) = 5√3/2 cm ≈ 4.33 cm

Example 5: Finding the Height of an Object

Problem: From a point on the ground, the angle of elevation to the top of a building is 60°. If you are standing 20 meters from the base of the building, how tall is it?

Solution:

1. sin 60° = height/distance
2. height = distance × sin 60°
3. height = 20 × (√3/2)
4. height = 10√3 meters ≈ 17.32 meters

Example 6: Vector Components

Problem: A force of 100 N acts at an angle of 60° with the horizontal. Find its vertical component.

Solution:

1. Vertical component = Force × sin 60°
2. Vertical component = 100 × (√3/2)
3. Vertical component = 50√3 N ≈ 86.6 N

Sin 60 in the Unit Circle

The sin 60 value plays a significant role in the unit circle:

• At 60° (or π/3 radians), the point on the unit circle is (1/2, √3/2)
• The y-coordinate of this point is sin 60° = √3/2
• This is one of the "special angles" that have exact values

Relationship of Sin 60 to Other Trigonometric Values

The sin 60 value is related to other trigonometric values:

1. sin 60° = cos 30° (complementary angles)
2. sin 60° = sin(180° - 60°) = sin 120° (supplementary angles)
3. sin² 60° + cos² 60° = (√3/2)² + (1/2)² = 3/4 + 1/4 = 1 (Pythagorean identity)

Memorizing Sin 60

To remember the sin 60 value, note these patterns:

• For special angles (30°, 45°, 60°), the sine values increase: 1/2, √2/2, √3/2
• The value √3/2 ≈ 0.866 is close to 0.9, which is close to 1
• This makes sense as 60° is approaching 90° (where sin 90° = 1)

Real-World Applications of Sin 60

The sin 60 value appears in many practical applications:

1. Engineering: Calculating forces on inclined surfaces at 60° angles
2. Architecture: Designing roof trusses with 60° angles
3. Navigation: Determining distances when angles of elevation are 60°
4. Physics: Analyzing components of vectors at 60° angles
5. Optics: Calculating refraction when light hits surfaces at 60° angles
6. Music theory: The sine wave at 60° phase shift in sound synthesis
7. Surveying: Measuring heights and distances using 60° angles
8. Crystallography: Many crystals have structures based on 60° angles

Practice Problems Using Sin 60

Test your understanding of the sin 60 value with these problems:

1. Find the area of an equilateral triangle with sides of length 12 cm.
2. A ladder 10 meters long leans against a wall at an angle of 60° with the ground. How high does the ladder reach on the wall?
3. Find the horizontal and vertical components of a 50 N force acting at an angle of 60° with the horizontal.
4. The angle of elevation from a point on the ground to the top of a tree is 60°. If the point is 15 meters from the base of the tree, find the height of the tree.

Sin 60 on the Calculator

To find sin 60 on a calculator:

1. Make sure your calculator is in degree mode (not radian mode)
2. Enter 60
3. Press the sin button
4. The result should be approximately 0.866025...

Conclusion

The sin 60 value (√3/2 or approximately 0.866) is one of the fundamental values in trigonometry. Understanding this value and knowing how to apply it in various contexts is essential for success in mathematics, physics, engineering, and many other fields.

Remember that sin 60 equals √3/2 exactly, and this precise value should be used in calculations whenever possible for accuracy. The decimal approximation of 0.866 is useful for quick estimations.

By practicing with the examples and problems presented in this guide, you'll develop proficiency in working with the sin 60 value in different trigonometric applications