What are Trigonometric Ratios for 90 Degrees?

S6-SA2-0019

Grade Level:

Class 10

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

Definition

What is it?

Trigonometric ratios for 90 degrees tell us the values of sine, cosine, and tangent when the angle in a right-angled triangle approaches 90 degrees. Since a triangle cannot have two 90-degree angles, we imagine what happens as one angle gets very, very close to 90 degrees.

Simple Example

Quick Example

Imagine a ladder leaning against a wall. If the ladder is almost straight up, making a very small angle with the wall, the angle it makes with the ground is almost 90 degrees. At this point, the 'base' of the triangle (distance from wall to ladder's foot) becomes tiny, and the 'height' (how high the ladder reaches) becomes almost equal to the ladder's length.

Worked Example

Step-by-Step

Let's find sin(90 degrees).
---We know sin(theta) = Opposite / Hypotenuse.
---Imagine a right-angled triangle PQR, with angle Q = 90 degrees and angle R = theta. So angle P = 90 - theta.
---As theta approaches 90 degrees, angle P approaches 0 degrees.
---When angle P becomes almost 0, the side PQ (opposite to R) becomes almost equal to the hypotenuse PR. The side QR (adjacent to R) becomes very, very small, almost zero.
---So, Opposite side (PQ) becomes almost equal to Hypotenuse (PR).
---Therefore, sin(90 degrees) = PQ / PR = 1 (since PQ is almost equal to PR).
---So, sin(90 degrees) = 1.

Why It Matters

Understanding these ratios is crucial for engineers designing buildings or bridges, as it helps calculate forces and stability. Space scientists at ISRO use these to track satellite paths and rocket trajectories. Even in game development, these ratios help create realistic movements and angles for characters and objects.

Common Mistakes

MISTAKE: Thinking tan(90) is 1 because sin(90) is 1 and cos(90) is 0. | CORRECTION: tan(theta) = sin(theta) / cos(theta). If cos(theta) is 0, division by zero is undefined, so tan(90) is undefined.

MISTAKE: Confusing the values of sin(90) and cos(90). | CORRECTION: Remember sin(90) = 1 and cos(90) = 0. A simple way to remember is that sine starts at 0 for 0 degrees and goes up to 1 for 90 degrees, while cosine starts at 1 for 0 degrees and goes down to 0 for 90 degrees.

MISTAKE: Trying to draw a physical triangle with a 90-degree angle for the 'theta' angle. | CORRECTION: Remember we are considering what happens as the angle 'approaches' 90 degrees, not actually 'is' 90 degrees in the context of the angle 'theta' itself within a right triangle.

Practice Questions

Try It Yourself

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

QUESTION: If sin(A) = 1, what is the value of angle A in degrees? | ANSWER: 90 degrees

QUESTION: Find the value of (sin(90) + cos(90))^2. | ANSWER: (1 + 0)^2 = 1^2 = 1

MCQ

Quick Quiz

Which of the following trigonometric ratios for 90 degrees is undefined?

sin(90)

cos(90)

tan(90)

None of the above

The Correct Answer Is:

tan(90) is undefined because it is sin(90)/cos(90) = 1/0, and division by zero is not allowed. Sin(90) is 1 and cos(90) is 0.

Real World Connection

In the Real World

When a drone takes off straight up, its path makes a 90-degree angle with the ground. Understanding sin(90) = 1 helps engineers calculate the maximum vertical lift possible, which is essential for drone delivery services like those planned in remote Indian areas.

Key Vocabulary

Key Terms

SINE: Ratio of opposite side to hypotenuse | COSINE: Ratio of adjacent side to hypotenuse | TANGENT: Ratio of opposite side to adjacent side | UNDEFINED: A value that cannot be determined, usually due to division by zero.

What's Next

What to Learn Next

Now that you understand ratios for 90 degrees, you're ready to explore trigonometric ratios for 0 degrees and 180 degrees. These will help you complete your understanding of angles at the boundaries and are super useful in higher math!