What is the Value of tan 90?

S6-SA2-0052

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 is 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 drawing a cricket pitch. If you try to calculate the slope of a perfectly vertical boundary rope (like a wall), you'll find it's impossible to give a single number. This is similar to how tan 90 is undefined; it represents an 'infinite' steepness or vertical line.

Worked Example

Step-by-Step

Let's find the value of tan 90 degrees using the definition of tangent in a right-angled triangle. --- Step 1: Recall the definition of tangent for an angle in a right-angled triangle: tan(theta) = Opposite side / Adjacent side. --- Step 2: Consider a right-angled triangle where one angle, theta, approaches 90 degrees. As theta gets closer to 90 degrees, the adjacent side to theta becomes very, very small, almost zero. --- Step 3: At exactly 90 degrees, the adjacent side becomes zero. --- Step 4: So, tan 90 = Opposite side / 0. --- Step 5: Division by zero is undefined in mathematics. --- 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 avoiding undefined scenarios is key for safe launches. It's also vital for careers in AI/ML where algorithms need to handle all possible inputs without crashing.

Common Mistakes

MISTAKE: Thinking tan 90 is 0 or 1. | CORRECTION: Tan 90 is undefined, not a specific number like 0 or 1. Remember sin 90 is 1 and cos 90 is 0.

MISTAKE: Confusing tan 90 with tan 0. | CORRECTION: Tan 0 is 0 (since sin 0 = 0 and cos 0 = 1). Tan 90 is undefined, they are very different.

MISTAKE: Forgetting that division by zero makes a value undefined. | CORRECTION: Always remember the rule: any number divided by zero is undefined. Since tan 90 = sin 90 / cos 90 = 1 / 0, it is undefined.

Practice Questions

Try It Yourself

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

QUESTION: If sin x = 1 and cos x = 0, what is the value of tan x? | ANSWER: tan x = sin x / cos x = 1 / 0, which is undefined.

QUESTION: A ramp is designed to be perfectly vertical. If you try to calculate its slope using the tangent function, what value would you get? | ANSWER: Undefined, because a perfectly vertical line has an undefined slope, just like tan 90.

MCQ

Quick Quiz

Which of the following describes the value of tan 90 degrees?

Undefined

Infinity

The Correct Answer Is:

The correct answer is Undefined. This is because tan 90 is calculated as sin 90 / cos 90, which is 1 / 0. Division by zero is always undefined in mathematics.

Real World Connection

In the Real World

In navigation systems like those used by auto-rickshaw drivers or food delivery apps, GPS coordinates involve angles. While a 90-degree angle might seem simple, understanding its trigonometric properties, including undefined values, is crucial for programmers to ensure the app doesn't crash when calculating extreme directions or slopes on a map.

Key Vocabulary

Key Terms

TANGENT: The ratio of the opposite side to the adjacent side in a right-angled triangle for a given angle. | 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, typically due to division by zero.

What's Next

What to Learn Next

Now that you understand tan 90, explore the values of tangent for other special angles like 180, 270, and 360 degrees. This will help you fully grasp the periodic nature of trigonometric functions and how they behave across the entire unit circle!