What is the Derivation of sin²A + cos²A = 1?

S6-SA2-0219

Grade Level:

Class 10

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

Definition

What is it?

The derivation of sin²A + cos²A = 1 shows how this fundamental trigonometric identity comes directly from the Pythagorean theorem when applied to a right-angled triangle. It proves that the square of the sine of an angle plus the square of the cosine of the same angle always equals 1. This identity is a cornerstone for solving many trigonometry problems.

Simple Example

Quick Example

Imagine you're checking your cricket score. If a player scores 60 runs out of 100 balls, and hits 8 boundaries, these numbers relate to their performance. Similarly, in a right-angled triangle, the lengths of the sides are related by the Pythagorean theorem, and this relationship is what helps us derive sin²A + cos²A = 1.

Worked Example

Step-by-Step

Let's derive sin²A + cos²A = 1 using a right-angled triangle.

1. Draw a right-angled triangle ABC, with the right angle at B.
---2. Let angle A be the angle we are considering. The side opposite to angle A is BC (Perpendicular, P), the side adjacent to angle A is AB (Base, B), and the longest side opposite the right angle is AC (Hypotenuse, H).
---3. According to the Pythagorean theorem, in triangle ABC: AB² + BC² = AC².
---4. Now, recall the definitions of sine and cosine: sin A = Perpendicular/Hypotenuse = BC/AC and cos A = Base/Hypotenuse = AB/AC.
---5. Let's find sin²A and cos²A: sin²A = (BC/AC)² = BC²/AC² and cos²A = (AB/AC)² = AB²/AC².
---6. Add them together: sin²A + cos²A = BC²/AC² + AB²/AC² = (BC² + AB²)/AC².
---7. From step 3, we know BC² + AB² = AC². So, substitute this into the equation: (AC²)/AC².
---8. (AC²)/AC² simplifies to 1. Therefore, sin²A + cos²A = 1.

Answer: The identity sin²A + cos²A = 1 is derived directly from the Pythagorean theorem.

Why It Matters

This identity is super important in fields like Physics and Engineering for calculating forces, trajectories, and waves. It helps AI/ML algorithms understand patterns in data, and engineers use it to design everything from bridges to mobile phone antennas. Knowing this helps you build a strong foundation for future studies in science and technology.

Common Mistakes

MISTAKE: Writing sinA² + cosA² = 1 | CORRECTION: The square applies to the entire sine or cosine value, not just the angle. It should be (sin A)² or sin²A, and (cos A)² or cos²A.

MISTAKE: Assuming sin²A + cos²B = 1 for different angles A and B | CORRECTION: The identity sin²A + cos²A = 1 is only true when both sine and cosine refer to the *same* angle, 'A'.

MISTAKE: Confusing sin²A with sin(A²) | CORRECTION: sin²A means (sin A) * (sin A). sin(A²) means finding the sine of the angle A squared, which is a very different value.

Practice Questions

Try It Yourself

QUESTION: If sin A = 3/5, find cos A using the identity sin²A + cos²A = 1. | ANSWER: cos A = 4/5

QUESTION: A right-angled triangle has sides 7 cm, 24 cm, and 25 cm. If A is the angle opposite the 7 cm side, verify that sin²A + cos²A = 1. | ANSWER: sin A = 7/25, cos A = 24/25. (7/25)² + (24/25)² = 49/625 + 576/625 = 625/625 = 1. Verified.

QUESTION: Simplify the expression: (1 - cos²θ) / sin²θ. | ANSWER: 1 (Since 1 - cos²θ = sin²θ, the expression becomes sin²θ / sin²θ = 1)

MCQ

Quick Quiz

Which theorem is directly used to derive the identity sin²A + cos²A = 1?

Thales's Theorem

Pythagorean Theorem

Midpoint Theorem

Basic Proportionality Theorem

The Correct Answer Is:

The identity sin²A + cos²A = 1 is derived by applying the Pythagorean theorem (a² + b² = c²) to the sides of a right-angled triangle and then using the definitions of sine and cosine.

Real World Connection

In the Real World

Imagine engineers at ISRO designing a new satellite. They use trigonometry, including identities like sin²A + cos²A = 1, to calculate the precise angles and distances for the satellite's trajectory in space. This ensures it reaches its orbit correctly, like how Chandrayaan-3 successfully landed on the moon!

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all values of the variables involved | PYTHAGOREAN THEOREM: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides | SINE (sin): Ratio of the opposite side to the hypotenuse in a right-angled triangle | COSINE (cos): Ratio of the adjacent side to the hypotenuse in a right-angled triangle | RIGHT-ANGLED TRIANGLE: A triangle with one angle measuring 90 degrees.

What's Next

What to Learn Next

Now that you understand this fundamental identity, you're ready to explore other trigonometric identities like 1 + tan²A = sec²A and 1 + cot²A = cosec²A. These identities also stem from the Pythagorean theorem and will help you solve even more complex trigonometry problems!