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What is the Pythagorean Identity sin²A + cos²A = 1?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The Pythagorean Identity, sin²A + cos²A = 1, is a fundamental rule in trigonometry. It states that for any angle 'A' in a right-angled triangle, if you square the sine of that angle and add it to the square of the cosine of the same angle, the result will always be 1. It's like a special 'Pythagoras theorem' for trigonometric ratios.
Simple Example
Quick Example
Imagine you have a cricket bat and a ball. If you know the 'swing angle' (A) and can find its sine and cosine values, this identity tells you that (sin A * sin A) + (cos A * cos A) will always be 1. It's a constant relationship, no matter how hard or softly you hit, as long as the angle A is the same.
Worked Example
Step-by-Step
Let's check the identity for an angle A = 30 degrees.
Step 1: Find sin(30 degrees). We know sin(30 degrees) = 1/2.
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Step 2: Find cos(30 degrees). We know cos(30 degrees) = sqrt(3)/2.
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Step 3: Calculate sin²(30 degrees). sin²(30 degrees) = (1/2)² = 1/4.
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Step 4: Calculate cos²(30 degrees). cos²(30 degrees) = (sqrt(3)/2)² = 3/4.
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Step 5: Add the results from Step 3 and Step 4. sin²(30 degrees) + cos²(30 degrees) = 1/4 + 3/4.
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Step 6: Simplify the sum. 1/4 + 3/4 = 4/4 = 1.
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Answer: Since 1 = 1, the identity sin²A + cos²A = 1 holds true for A = 30 degrees.
Why It Matters
This identity is super important for simplifying complex math problems in engineering and physics. It helps scientists at ISRO calculate trajectories for satellites and engineers design bridges. Understanding it can open doors to careers in Space Technology, AI, and even game development!
Common Mistakes
MISTAKE: Writing sinA² instead of sin²A. | CORRECTION: sin²A means (sin A) * (sin A). sinA² means sin(A*A), which is very different. Always remember the square is on 'sin' or 'cos', not on 'A'.
MISTAKE: Thinking the identity works for any triangle, not just right-angled triangles (where sin and cos are defined). | CORRECTION: The basic definitions of sine and cosine come from right-angled triangles. While it extends to other triangles using different laws, the direct derivation of sin²A + cos²A = 1 is rooted in right-angled triangle properties.
MISTAKE: Forgetting that the angle 'A' must be the same for both sin and cos in the identity. | CORRECTION: The identity is sin²A + cos²A = 1, not sin²A + cos²B = 1. The angle must be identical for both terms.
Practice Questions
Try It Yourself
QUESTION: If sin A = 3/5, find cos A using the Pythagorean Identity. | ANSWER: cos A = 4/5
QUESTION: Simplify the expression: 5 sin²A + 5 cos²A. | ANSWER: 5
QUESTION: If cos A = 0.6, find the value of sin A. | ANSWER: sin A = 0.8
MCQ
Quick Quiz
Which of the following is equivalent to 1 - cos²A?
sin A
cos A
sin²A
tan²A
The Correct Answer Is:
C
The Pythagorean Identity is sin²A + cos²A = 1. If we rearrange it, we get sin²A = 1 - cos²A. So, option C is correct.
Real World Connection
In the Real World
This identity is crucial in Computer Graphics, like those used in games or movie animation. When a character moves, its position and orientation are often calculated using trigonometry. The Pythagorean Identity helps ensure that movements are smooth and realistic, preventing glitches and making virtual worlds look believable.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all possible values of its variables. | TRIGONOMETRY: The branch of mathematics dealing with the relationships between the sides and angles of triangles. | SINE: Ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. | COSINE: Ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. | ANGLE: The space between two intersecting lines or surfaces at or close to the point where they meet.
What's Next
What to Learn Next
Now that you understand this basic identity, you're ready to explore other trigonometric identities like tan²A + 1 = sec²A and 1 + cot²A = cosec²A. These build on the same principles and will help you solve even more complex problems in higher classes.
