What is the Value of sin 60?

S6-SA2-0047

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The value of sin 60 degrees is a specific ratio of sides in a right-angled triangle. It represents the ratio of the length of the side opposite the 60-degree angle to the length of the hypotenuse. This value is a constant, always sqrt(3)/2.

Simple Example

Quick Example

Imagine a ladder leaning against a wall, making a 60-degree angle with the ground. If the ladder is 2 meters long (hypotenuse), then the height it reaches up the wall (opposite side) would be 2 * sin 60. So, it reaches 2 * (sqrt(3)/2) = sqrt(3) meters up the wall.

Worked Example

Step-by-Step

Problem: Find the value of sin 60 degrees.
--- Step 1: Recall the special right-angled triangle with angles 30, 60, and 90 degrees.
--- Step 2: Remember the side ratios for this triangle: if the side opposite 30 degrees is 'x', then the side opposite 60 degrees is 'x * sqrt(3)', and the hypotenuse is '2x'.
--- Step 3: The definition of sin (sine) for an angle in a right-angled triangle is (Opposite side) / (Hypotenuse).
--- Step 4: For the 60-degree angle, the opposite side is 'x * sqrt(3)' and the hypotenuse is '2x'.
--- Step 5: Substitute these values into the sine definition: sin 60 = (x * sqrt(3)) / (2x).
--- Step 6: Cancel out 'x' from the numerator and denominator.
--- Answer: The value of sin 60 degrees is sqrt(3)/2.

Why It Matters

Understanding sin 60 is crucial for solving problems in Physics, like calculating projectile motion or forces, and in Engineering for designing structures. It's used by Space Technologists to plan satellite trajectories and by AI/ML engineers in complex mathematical models.

Common Mistakes

MISTAKE: Confusing sin 60 with cos 60 or tan 60 values. | CORRECTION: Always remember SOH CAH TOA. Sin is Opposite/Hypotenuse, Cos is Adjacent/Hypotenuse, Tan is Opposite/Adjacent.

MISTAKE: Writing sin 60 as 1/2 or 1/sqrt(2). | CORRECTION: The value of sin 60 is sqrt(3)/2. 1/2 is sin 30, and 1/sqrt(2) is sin 45.

MISTAKE: Not knowing how to derive the value using a 30-60-90 triangle. | CORRECTION: Practice drawing an equilateral triangle, bisecting one angle to create a 30-60-90 triangle, and labelling the sides to derive the ratios.

Practice Questions

Try It Yourself

QUESTION: What is the exact value of sin 60 degrees? | ANSWER: sqrt(3)/2

QUESTION: If a ramp makes a 60-degree angle with the ground and its length (hypotenuse) is 10 meters, what is the vertical height it reaches? (Use sin 60 = sqrt(3)/2) | ANSWER: 5 * sqrt(3) meters

QUESTION: A kite string is 50 meters long and makes an angle of 60 degrees with the horizontal ground. Assuming the string is straight, what is the approximate height of the kite above the ground? (Use sqrt(3) approx 1.732) | ANSWER: 50 * (sqrt(3)/2) = 25 * sqrt(3) approx 25 * 1.732 = 43.3 meters

MCQ

Quick Quiz

Which of the following is the correct value for sin 60 degrees?

2026-01-02T00:00:00.000Z

sqrt(3)/2

1/sqrt(2)

sqrt(3)

The Correct Answer Is:

The value of sin 60 degrees is the ratio of the opposite side to the hypotenuse in a 30-60-90 triangle, which is consistently sqrt(3)/2. Options A, C, and D are values for other angles or trigonometric functions.

Real World Connection

In the Real World

Engineers building flyovers and bridges in India often use trigonometric values like sin 60 to calculate the slopes and heights needed for construction. For example, when designing the angle of approach for a ramp, they ensure the vertical rise matches the horizontal distance using these values for safety and stability.

Key Vocabulary

Key Terms

SINE: The ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle. | OPPOSITE SIDE: The side across from a given angle in a triangle. | RIGHT-ANGLED TRIANGLE: A triangle with one angle measuring 90 degrees. | TRIGONOMETRY: The branch of mathematics dealing with the relations of the sides and angles of triangles.

What's Next

What to Learn Next

Great job understanding sin 60! Next, explore the values of cos 60 and tan 60. Knowing these will complete your understanding of trigonometric ratios for special angles and help you solve more complex problems in geometry and physics.