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What is the Value of sin 60 degrees?
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 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 fixed constant in trigonometry.
Simple Example
Quick Example
Imagine you're flying a kite. If the kite string makes an angle of 60 degrees with the ground, and you want to know how high the kite is (the opposite side) compared to the length of the string you've let out (the hypotenuse), sin 60 degrees helps you find that ratio. It's like knowing that for every 1 unit of string, the kite is sqrt(3)/2 units high.
Worked Example
Step-by-Step
Let's find the value of sin 60 degrees using an equilateral triangle.
1. Draw an equilateral triangle ABC with side length 2 units. All angles are 60 degrees.
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2. Draw an altitude AD from vertex A to side BC. This altitude bisects angle A (making angle BAD = 30 degrees) and side BC (making BD = 1 unit).
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3. Now, focus on the right-angled triangle ADB. Angle B is 60 degrees, angle BAD is 30 degrees, and angle ADB is 90 degrees.
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4. We know AB (hypotenuse) = 2 units and BD = 1 unit.
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5. Using the Pythagorean theorem in triangle ADB, AD^2 + BD^2 = AB^2. So, AD^2 + 1^2 = 2^2.
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6. AD^2 + 1 = 4, which means AD^2 = 3. So, AD = sqrt(3) units.
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7. In triangle ADB, sin 60 degrees (angle B) = Opposite side / Hypotenuse = AD / AB.
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8. Therefore, sin 60 degrees = sqrt(3) / 2.
Answer: sin 60 degrees = sqrt(3) / 2
Why It Matters
Understanding sin 60 degrees is crucial in fields like Physics to calculate forces and trajectories, and in Engineering for designing structures and bridges. It's used by scientists at ISRO to track satellite paths and by engineers building advanced AI systems for robotics, making it a foundational concept for many exciting careers.
Common Mistakes
MISTAKE: Confusing sin 60 with cos 60 or tan 60. | CORRECTION: Remember the SOH CAH TOA mnemonic: SOH means Sin = Opposite/Hypotenuse. For 60 degrees, the opposite side is sqrt(3) and hypotenuse is 2.
MISTAKE: Writing the value as 1/2 or 1/sqrt(2). | CORRECTION: The specific value for 60 degrees is sqrt(3)/2. 1/2 is for sin 30 degrees, and 1/sqrt(2) is for sin 45 degrees.
MISTAKE: Not rationalizing the denominator if the value was expressed differently (e.g., 3/sqrt(3)). | CORRECTION: While for sin 60, sqrt(3)/2 is already rationalized, always ensure the denominator does not have a square root when simplifying other trigonometric values.
Practice Questions
Try It Yourself
QUESTION: What is the exact value of sin 60 degrees? | ANSWER: sqrt(3)/2
QUESTION: If a ladder leans against a wall making a 60-degree angle with the ground, and the ladder is 4 meters long, how high up the wall does it reach? (Hint: Use sin 60 degrees) | ANSWER: 2 * sqrt(3) meters
QUESTION: An equilateral triangle has a side length of 6 cm. What is the length of its altitude? (Hint: The altitude forms a right triangle with a 60-degree angle and half the base). | ANSWER: 3 * sqrt(3) cm
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:
B
The value of sin 60 degrees is sqrt(3)/2. This comes from the ratio of the opposite side to the hypotenuse in a right-angled triangle with a 60-degree angle. Options A, C, and D are values for other angles or functions.
Real World Connection
In the Real World
Imagine an architect designing a ramp for a building or a road. If the ramp needs to rise at a specific angle, say 60 degrees, they use trigonometry, including sin 60 degrees, to calculate how long the ramp needs to be for a certain height, ensuring it's safe and functional. This is vital in construction projects across India.
Key Vocabulary
Key Terms
TRIGONOMETRY: The branch of mathematics dealing with the relations between the sides and angles of triangles | RIGHT-ANGLED TRIANGLE: A triangle with one angle measuring 90 degrees | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle | OPPOSITE SIDE: The side across from a given angle in a right-angled triangle | ALTITUDE: A line segment from a vertex perpendicular to the opposite side.
What's Next
What to Learn Next
Great job learning about sin 60 degrees! Next, you should explore the values of cos 60 degrees and tan 60 degrees. Understanding these will complete your knowledge of trigonometric ratios for this important angle and open doors to solving more complex problems.
