What is tan 90 degrees? | Simple Explanation for Class 10

S6-SA2-0300

What is the Value of tan 90 degrees?

Grade Level:

Class 10

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

Definition

What is it?

The value of tan 90 degrees is undefined. This happens because the tangent function is defined as the ratio of the sine of an angle to its cosine, and the cosine of 90 degrees is zero, leading to division by zero.

Simple Example

Quick Example

Imagine you are trying to share a pack of 10 laddoos among 0 friends. You can't do it, right? It's an impossible task. Similarly, in trigonometry, when we try to calculate tan 90 degrees, we face a situation like dividing by zero, which is not possible.

Worked Example

Step-by-Step

Let's find the value of tan 90 degrees using its definition.
1. Recall the definition of the tangent function: tan(theta) = sin(theta) / cos(theta).
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2. Substitute theta = 90 degrees into the formula: tan 90 = sin 90 / cos 90.
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3. Remember the standard trigonometric values for 90 degrees: sin 90 = 1 and cos 90 = 0.
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4. Substitute these values into the equation: tan 90 = 1 / 0.
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5. Division by zero is mathematically undefined.
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ANSWER: Therefore, the value of tan 90 degrees is Undefined.

Why It Matters

Understanding undefined values is crucial in fields like Engineering and Physics, especially when dealing with limits or singularities. For example, in Space Technology, calculating rocket trajectories involves complex angles and trigonometric functions, where knowing when a value becomes undefined can prevent critical errors. It's also important for careers in AI/ML, where algorithms might encounter division by zero scenarios.

Common Mistakes

MISTAKE: Thinking tan 90 is 0 or 1. | CORRECTION: tan 90 is undefined, not 0 or 1. Remember sin 90 = 1 and cos 90 = 0.

MISTAKE: Confusing tan 0 with tan 90. | CORRECTION: tan 0 = sin 0 / cos 0 = 0 / 1 = 0. tan 90 = sin 90 / cos 90 = 1 / 0, which is undefined.

MISTAKE: Assuming undefined means 'very large number'. | CORRECTION: While it approaches a very large number as the angle approaches 90, at exactly 90 degrees, it is precisely 'undefined' because division by zero is not allowed.

Practice Questions

Try It Yourself

QUESTION: What is the value of cos 90 degrees? | ANSWER: 0

QUESTION: If tan(theta) = sin(theta) / cos(theta), and cos(theta) = 0, what can you say about tan(theta)? | ANSWER: tan(theta) will be undefined.

QUESTION: A right-angled triangle has an angle that approaches 90 degrees. As this angle gets closer to 90 degrees, what happens to the ratio of the opposite side to the adjacent side? | ANSWER: The ratio (which is tan of the angle) becomes very, very large, eventually becoming undefined at exactly 90 degrees.

MCQ

Quick Quiz

Which of the following statements about tan 90 degrees is true?

Its value is 0.

Its value is 1.

Its value is undefined.

Its value is -1.

The Correct Answer Is:

tan 90 degrees is calculated as sin 90 / cos 90. Since sin 90 = 1 and cos 90 = 0, we get 1/0, which is undefined. Options A, B, and D are incorrect.

Real World Connection

In the Real World

Imagine you are an engineer designing a mobile tower. You use trigonometry to calculate heights and distances. If an angle in your calculations leads to tan 90, it means your design might have a vertical component that is infinitely long or a horizontal component that is zero, indicating a potential instability or an impossible scenario in your structure. It's like trying to make a perfectly vertical wall with no base, which is impossible in the real world.

Key Vocabulary

Key Terms

TANGENT: The ratio of the opposite side to the adjacent side in a right-angled triangle, or sin(theta)/cos(theta). | SINE: The ratio of the opposite side to the hypotenuse. | COSINE: The ratio of the adjacent side to the hypotenuse. | UNDEFINED: A mathematical expression that does not have a meaningful value, often due to division by zero.

What's Next

What to Learn Next

Now that you understand undefined values, next you can explore the graphs of trigonometric functions. Seeing how the graph of tan(x) behaves as x approaches 90 degrees will give you a deeper visual understanding of why it becomes undefined. Keep up the great work!