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What is the Pythagorean Identity sin^2 A + cos^2 A = 1?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The Pythagorean Identity, sin^2 A + cos^2 A = 1, is a fundamental rule in trigonometry. It states that for any angle 'A' in a right-angled triangle, the square of the sine of the angle plus the square of the cosine of the angle will always add up to 1. This identity comes directly from the famous Pythagorean Theorem.
Simple Example
Quick Example
Imagine you're watching a cricket match, and a bowler throws a ball. The path of the ball, the ground, and the height it reaches can form a right-angled triangle. If you know the 'sin' and 'cos' values for the angle the ball makes with the ground, this identity tells you that (sin of that angle)^2 + (cos of that angle)^2 will always be 1. It's like a guaranteed score!
Worked Example
Step-by-Step
Let's check the identity for an angle of 30 degrees.
Step 1: Recall the values for sin 30 degrees and cos 30 degrees.
---sin 30 degrees = 1/2
---cos 30 degrees = sqrt(3)/2
Step 2: Calculate sin^2 30 degrees.
---sin^2 30 degrees = (1/2)^2 = 1/4
Step 3: Calculate cos^2 30 degrees.
---cos^2 30 degrees = (sqrt(3)/2)^2 = 3/4
Step 4: Add the results from Step 2 and Step 3.
---sin^2 30 degrees + cos^2 30 degrees = 1/4 + 3/4 = 4/4 = 1
Answer: As you can see, for 30 degrees, sin^2 A + cos^2 A indeed equals 1.
Why It Matters
This identity is super important for solving complex problems in many fields! Engineers use it to design safe buildings and bridges, and scientists use it in physics to understand waves and signals. It's also crucial in computer graphics for making realistic 3D animations and in space technology for calculating satellite orbits.
Common Mistakes
MISTAKE: Writing sin A^2 instead of (sin A)^2 | CORRECTION: Remember that sin^2 A means (sin A) * (sin A), not sin of (A*A). Always square the *value* of sin A, not the angle itself.
MISTAKE: Thinking sin^2 A + cos^2 B = 1 is true for different angles A and B. | CORRECTION: The identity is only true when both sine and cosine are taken for the *same* angle, 'A'.
MISTAKE: Confusing sin^2 A + cos^2 A = 1 with other identities like tan A = sin A / cos A. | CORRECTION: While all are trigonometric identities, they are distinct. This specific identity involves the sum of squares of sine and cosine.
Practice Questions
Try It Yourself
QUESTION: If sin A = 0.6, find cos A using the Pythagorean Identity. | ANSWER: cos A = 0.8
QUESTION: If 5 sin A = 3, show that 5 cos A = 4. (Hint: Use the identity to find cos A first). | ANSWER: sin A = 3/5. Using sin^2 A + cos^2 A = 1, (3/5)^2 + cos^2 A = 1. 9/25 + cos^2 A = 1. cos^2 A = 1 - 9/25 = 16/25. cos A = sqrt(16/25) = 4/5. So, 5 cos A = 5 * (4/5) = 4.
QUESTION: Simplify the expression (1 - sin^2 A) / cos^2 A. | ANSWER: 1
MCQ
Quick Quiz
Which of the following statements is always true?
sin A + cos A = 1
sin^2 A - cos^2 A = 1
sin^2 A + cos^2 A = 1
sin A * cos A = 1
The Correct Answer Is:
C
Option C, sin^2 A + cos^2 A = 1, is the correct Pythagorean Identity. Options A, B, and D are not generally true for all angles.
Real World Connection
In the Real World
This identity is used in GPS systems in your mobile phone! When your phone calculates your location, it uses signals from satellites. These calculations often involve trigonometry to pinpoint your exact spot on Earth, and the Pythagorean Identity helps simplify these complex position equations, making sure your Ola or Swiggy delivery reaches you accurately.
Key Vocabulary
Key Terms
IDENTITY: A mathematical equation that is true for all possible values of the variables. | SINE (sin): A trigonometric ratio in a right-angled triangle, opposite side / hypotenuse. | COSINE (cos): A trigonometric ratio in a right-angled triangle, adjacent side / hypotenuse. | PYTHAGOREAN THEOREM: In a right-angled triangle, a^2 + b^2 = c^2, where c is the hypotenuse.
What's Next
What to Learn Next
Great job understanding this core identity! Next, you should explore other important trigonometric identities like 1 + tan^2 A = sec^2 A and 1 + cot^2 A = cosec^2 A. They build directly on this one and will unlock even more ways to solve trigonometry problems.
