What is the Derivation of sin^2 A + cos^2 A = 1?

S6-SA2-0058

Grade Level:

Class 10

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

Definition

What is it?

The derivation of sin^2 A + cos^2 A = 1 shows how this fundamental trigonometric identity comes directly from the Pythagorean theorem applied to a right-angled triangle. It proves that for any angle A in a right triangle, the square of its sine plus the square of its cosine always equals 1. This identity is a cornerstone of trigonometry, making complex calculations simpler.

Simple Example

Quick Example

Imagine you have a ladder leaning against a wall, forming a right-angled triangle with the ground. If the ladder is 5 meters long, and it reaches a height of 3 meters on the wall, the distance from the wall to the ladder's base on the ground would be 4 meters (using Pythagoras: 3^2 + 4^2 = 5^2). This identity is like a shortcut that tells us that the 'angle relationship' between the ladder, wall, and ground will always follow sin^2 A + cos^2 A = 1, no matter how steep the ladder is.

Worked Example

Step-by-Step

Let's derive sin^2 A + cos^2 A = 1 using a right-angled triangle.

Step 1: Consider a right-angled triangle ABC, with the right angle at B. Let angle C be 'A'.

Step 2: According to the Pythagorean theorem, in triangle ABC, we have AB^2 + BC^2 = AC^2. (Here, AC is the hypotenuse).

Step 3: Recall the definitions of sine and cosine for angle A:
sin A = Opposite side / Hypotenuse = AB / AC
cos A = Adjacent side / Hypotenuse = BC / AC

Step 4: Square both sin A and cos A:
sin^2 A = (AB / AC)^2 = AB^2 / AC^2
cos^2 A = (BC / AC)^2 = BC^2 / AC^2

Step 5: Add sin^2 A and cos^2 A:
sin^2 A + cos^2 A = (AB^2 / AC^2) + (BC^2 / AC^2)

Step 6: Combine the fractions since they have the same denominator:
sin^2 A + cos^2 A = (AB^2 + BC^2) / AC^2

Step 7: From Step 2, we know that AB^2 + BC^2 = AC^2. Substitute this into the equation:
sin^2 A + cos^2 A = AC^2 / AC^2

Step 8: Since AC^2 / AC^2 = 1, we get:
sin^2 A + cos^2 A = 1

This proves the identity. It holds true for any angle A in a right-angled triangle.

Why It Matters

This identity is crucial for simplifying complex trigonometric expressions in fields like engineering and physics. Engineers use it to design bridges and buildings, ensuring stability. It's also vital in AI/ML for understanding data patterns and in space technology for calculating satellite trajectories.

Common Mistakes

MISTAKE: Writing sin A^2 instead of sin^2 A. | CORRECTION: sin^2 A means (sin A) * (sin A). sin A^2 means sin of (A squared), which is very different.

MISTAKE: Assuming sin^2 A + cos^2 B = 1. | CORRECTION: The angles for both sine and cosine must be the same (sin^2 A + cos^2 A = 1) for the identity to hold true.

MISTAKE: Forgetting that this identity comes from the Pythagorean theorem. | CORRECTION: Always remember the link to right-angled triangles and Pythagoras (Opposite^2 + Adjacent^2 = Hypotenuse^2) to understand its origin.

Practice Questions

Try It Yourself

QUESTION: If sin A = 3/5, what is cos^2 A? | ANSWER: cos^2 A = 1 - sin^2 A = 1 - (3/5)^2 = 1 - 9/25 = 16/25.

QUESTION: Simplify: (1 - sin^2 A) / cos^2 A | ANSWER: (1 - sin^2 A) / cos^2 A = cos^2 A / cos^2 A = 1.

QUESTION: If 5 sin A = 4, find the value of (cos A - sin A) / (cos A + sin A). | ANSWER: From 5 sin A = 4, sin A = 4/5. Using sin^2 A + cos^2 A = 1, (4/5)^2 + cos^2 A = 1, so 16/25 + cos^2 A = 1, which means cos^2 A = 9/25. Thus, cos A = 3/5 (assuming A is acute). Now substitute: (3/5 - 4/5) / (3/5 + 4/5) = (-1/5) / (7/5) = -1/7.

MCQ

Quick Quiz

Which of the following is equivalent to 1 - sin^2 A?

tan^2 A

cos^2 A

sec^2 A

cosec^2 A

The Correct Answer Is:

The fundamental identity is sin^2 A + cos^2 A = 1. Rearranging this, we get cos^2 A = 1 - sin^2 A. Therefore, option B is correct.

Real World Connection

In the Real World

This identity is used in mobile phone GPS systems. When your phone calculates your location, it uses signals from satellites. These signals involve trigonometric calculations to determine distances and angles, and identities like sin^2 A + cos^2 A = 1 help simplify these complex geometric problems, ensuring your food delivery driver finds you easily!

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: Ratio of the opposite side to the hypotenuse in a right triangle | COSINE: Ratio of the adjacent side to the hypotenuse in a right triangle | HYPOTENUSE: The longest side of a right-angled triangle, opposite the right angle

What's Next

What to Learn Next

Next, explore other important trigonometric identities like 1 + tan^2 A = sec^2 A and 1 + cot^2 A = cosec^2 A. Understanding these will further strengthen your ability to solve complex problems in trigonometry and prepare you for higher-level mathematics.