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What is the Value of tan 60 degrees?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The value of tan 60 degrees is a specific ratio in a right-angled triangle. It is the ratio of the length of the side opposite the 60-degree angle to the length of the side adjacent to the 60-degree angle. This value is a fixed constant, approximately 1.732.
Simple Example
Quick Example
Imagine a ladder leaning against a wall, making a 60-degree angle with the ground. If you know the height the ladder reaches on the wall (opposite side) and the distance of the ladder's base from the wall (adjacent side), the ratio of these two lengths will always be the value of tan 60 degrees. It's like a fixed 'slope factor' for that angle.
Worked Example
Step-by-Step
Let's find the value of tan 60 degrees using a special right-angled triangle.
1. Start with an equilateral triangle ABC, where all sides are equal, say 2 units each. All angles are 60 degrees.
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2. Draw a perpendicular AD from vertex A to side BC. This line AD bisects BC and also bisects angle A.
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3. Now, consider the right-angled triangle ADC. Angle C is 60 degrees, angle DAC is 30 degrees, and angle ADC is 90 degrees.
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4. In triangle ADC, the hypotenuse AC is 2 units. The side DC is half of BC, so DC = 1 unit.
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5. Using the Pythagorean theorem (a^2 + b^2 = c^2) in triangle ADC, we can find AD: AD^2 + DC^2 = AC^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. Now, for angle C (60 degrees) in triangle ADC:
Opposite side = AD = sqrt(3)
Adjacent side = DC = 1
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8. tan 60 degrees = (Opposite side) / (Adjacent side) = sqrt(3) / 1 = sqrt(3).
Therefore, the value of tan 60 degrees is sqrt(3).
Why It Matters
Understanding trigonometric values like tan 60 degrees is crucial in fields like Engineering and Physics to calculate forces, distances, and angles in structures. In Space Technology, it helps determine rocket trajectories and satellite positions. Even in AI/ML, these concepts are foundational for understanding vector rotations and transformations in data analysis.
Common Mistakes
MISTAKE: Confusing tan with sin or cos, or mixing up the ratios (opposite/hypotenuse, adjacent/hypotenuse). | CORRECTION: Remember 'SOH CAH TOA' - Tan is always Opposite/Adjacent. Practice drawing right triangles and labeling sides.
MISTAKE: Forgetting the exact value of tan 60 degrees and guessing. | CORRECTION: Memorize the special angle values for 0, 30, 45, 60, and 90 degrees. You can also quickly derive it using an equilateral triangle.
MISTAKE: Using the wrong angle in a multi-step problem, or assuming the angle given is always the one to use for tan. | CORRECTION: Always identify the correct angle in the right-angled triangle for which you need the tangent ratio. Double-check the problem statement.
Practice Questions
Try It Yourself
QUESTION: In a right-angled triangle, if the angle is 60 degrees, and the side opposite to it is 5 * sqrt(3) cm, what is the length of the side adjacent to the angle? | ANSWER: 5 cm
QUESTION: A tower casts a shadow 10 meters long when the angle of elevation of the sun is 60 degrees. What is the height of the tower? (Hint: The angle of elevation is the angle formed by the ground and the line of sight to the top of the tower). | ANSWER: 10 * sqrt(3) meters
QUESTION: If tan(x) = sqrt(3), what is the value of x (in degrees)? | ANSWER: 60 degrees
MCQ
Quick Quiz
What is the exact value of tan 60 degrees?
1/sqrt(3)
sqrt(3)
1
2
The Correct Answer Is:
B
The value of tan 60 degrees is sqrt(3). This can be derived from an equilateral triangle or remembered as a standard trigonometric ratio for special angles.
Real World Connection
In the Real World
When civil engineers design ramps for wheelchair access or flyovers in cities like Bengaluru or Mumbai, they use trigonometry to calculate the correct slope. If a ramp needs a specific incline, say related to a 60-degree angle for stability or height, knowing tan 60 degrees helps them determine the length of the ramp needed for a certain vertical rise, ensuring it meets safety standards.
Key Vocabulary
Key Terms
Tangent: The ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle for a given acute angle. | Right-angled triangle: A triangle with one angle measuring 90 degrees. | Opposite side: The side across from a given angle in a right-angled triangle. | Adjacent side: The side next to a given angle in a right-angled triangle, not the hypotenuse. | Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
What's Next
What to Learn Next
Great job understanding tan 60 degrees! Next, explore the values of tan for other special angles like 30 and 45 degrees, and also learn about sin and cos for these angles. This will complete your understanding of basic trigonometric ratios and set you up for solving more complex problems.
