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What is the Value of tan 60?
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 it. This value is a constant for any right-angled triangle with a 60-degree angle.
Simple Example
Quick Example
Imagine you are flying a kite and the string makes an angle of 60 degrees with the ground. If you want to know the ratio of the kite's height above the ground to how far horizontally it is from you, that ratio is tan 60. It helps you understand how 'steep' the kite string is.
Worked Example
Step-by-Step
To find the value of tan 60 degrees, we can use an equilateral triangle.
1. Draw an equilateral triangle ABC with side length 2 units. All angles are 60 degrees.
2. Draw an altitude AD from vertex A to side BC. This altitude bisects angle A (making it two 30-degree angles) and also bisects side BC (making BD = DC = 1 unit).
3. Now consider the right-angled triangle ADB. Angle B is 60 degrees, angle BAD is 30 degrees, and angle ADB is 90 degrees.
4. We know AB = 2 (hypotenuse) and BD = 1 (adjacent side to angle B).
5. Using Pythagoras theorem in triangle ADB: AD^2 + BD^2 = AB^2. So, AD^2 + 1^2 = 2^2.
6. AD^2 + 1 = 4, which means AD^2 = 3. So, AD = sqrt(3).
7. In triangle ADB, for angle B (60 degrees), the opposite side is AD = sqrt(3) and the adjacent side is BD = 1.
8. tan 60 = Opposite side / Adjacent side = AD / BD = sqrt(3) / 1 = sqrt(3).
Answer: The value of tan 60 degrees is sqrt(3).
Why It Matters
Understanding tan 60 helps engineers design ramps and bridges, ensuring stability and correct slopes. In space technology, ISRO scientists use trigonometry to calculate rocket trajectories and satellite orbits. This concept is fundamental for careers in architecture, civil engineering, and even game development.
Common Mistakes
MISTAKE: Confusing tan 60 with sin 60 or cos 60 | CORRECTION: Remember the SOH CAH TOA mnemonic: tan is Opposite/Adjacent. Always double-check the ratio you are using.
MISTAKE: Forgetting the exact value of tan 60 and guessing | CORRECTION: Either derive it using the 30-60-90 triangle method or memorize the standard trigonometric table for common angles.
MISTAKE: Using the wrong sides for Opposite and Adjacent in a right-angled triangle | CORRECTION: The 'Opposite' side is always across from the angle, and the 'Adjacent' side is next to the angle but not the hypotenuse.
Practice Questions
Try It Yourself
QUESTION: What is the approximate decimal value of tan 60? (Use sqrt(3) = 1.732) | ANSWER: 1.732
QUESTION: If in a right-angled triangle, the angle is 60 degrees and the side adjacent to it is 5 cm, what is the length of the side opposite to the 60-degree angle? | ANSWER: 5 * sqrt(3) cm or approximately 8.66 cm
QUESTION: An electric pole casts a shadow. If the angle of elevation of the sun is 60 degrees and the shadow is 10 meters long, what is the height of the pole? | ANSWER: 10 * sqrt(3) meters or approximately 17.32 meters
MCQ
Quick Quiz
Which of the following is the correct value of tan 60 degrees?
1/sqrt(3)
sqrt(3)/2
sqrt(3)
1
The Correct Answer Is:
C
tan 60 degrees is the ratio of the opposite side to the adjacent side in a 30-60-90 triangle, which simplifies to sqrt(3). Options A, B, and D are values for other trigonometric ratios or angles.
Real World Connection
In the Real World
When a crane lifts heavy construction materials, the angle its arm makes with the ground is crucial. Engineers use tan 60 (or other trigonometric values) to calculate the safe height the crane can reach and the stability required, preventing accidents at construction sites in Mumbai or Delhi.
Key Vocabulary
Key Terms
TRIGONOMETRY: The branch of mathematics dealing with the relations of the sides and angles of triangles | RIGHT-ANGLED TRIANGLE: A triangle with one angle measuring 90 degrees | OPPOSITE SIDE: The side across from a given angle in a right triangle | ADJACENT SIDE: The side next to a given angle (not the hypotenuse) | HYPOTENUSE: The longest side in a right-angled triangle, opposite the right angle
What's Next
What to Learn Next
Great job understanding tan 60! Next, explore the values of tan for other standard angles like 30 and 45 degrees, and then move on to understanding the concepts of sin and cos for these angles. This will build a strong foundation for advanced trigonometry.
