Pythagorean Identities
Learn the three fundamental Pythagorean identities and how to derive and apply them.
Definition
The Three Pythagorean Identities
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Worked Examples
Prove that using the unit circle.
Start with a point on the unit circle
Any point on the unit circle has coordinates →
Apply the unit circle equation
The unit circle is defined by →
Substitute the coordinates
Replace with and with →
Write in standard form
Use exponent notation: → ✓
Answer: The identity is proven by the definition of the unit circle!
Common Mistakes
Forgetting to consider the quadrant when taking square roots
Why it's wrong: The equation gives . Students often forget to determine which sign applies.
Correct: Always check which quadrant the angle is in to determine the sign of the trig function.
Writing as
Why it's wrong: means , not . The notation is shorthand for squaring the result of the sine function.
Correct: , meaning "find sine of theta, then square the result."
Applying an identity where a function is undefined
Why it's wrong: The identity is undefined when (at , etc.).
Correct: Check that the functions in the identity are defined for the given angle.
Interactive Visual
Unit Circle
0°
0
0
1
0
(1, 0)
Click and drag to rotate the angle around the unit circle.
Right Triangle Trigonometry
Remember: SOH-CAH-TOA
Sin = Opp / Hyp
Cos = Adj / Hyp
Tan = Opp / Adj
Click on sin, cos, or tan to highlight the relevant sides of the triangle.
Interactive Sandbox
Expression Calculator
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Practice Problems
15 problemsWhat is the fundamental Pythagorean identity?
Why It Matters
- Simplifying expressions: Convert between trig functions to simplify complex expressions
- Solving equations: Transform trigonometric equations into solvable forms
- Calculus: Used extensively in integration and differentiation
- Physics: Appear in wave equations, oscillations, and electromagnetic theory
- Verifying identities: Serve as building blocks for proving other trig identities
Real World Applications
Signal Processing
Engineers use Pythagorean identities when analyzing radio and audio signals that are modeled by sine and cosine waves.
Example:
When combining two signals and , the total power is , which is constant.
A signal has components and .
What is the total amplitude?
Step 1: Write the mathematical expression
Use to find the amplitude squared:
Physics: Simple Harmonic Motion
The position and velocity of an oscillating object are related through sine and cosine. Their energy relationship uses the Pythagorean identity.
Example:
If position is and velocity is , then .
A pendulum has position at some instant.
If the maximum position is , what fraction of is the velocity?
Step 1: Write the mathematical expression
Use with :
Key Takeaways
- 1The fundamental Pythagorean identity is
- 2Dividing by gives
- 3Dividing by gives
- 4These identities let you convert between trig functions and simplify expressions
- 5Always consider the quadrant when taking square roots
Frequently Asked Questions
Glossary
- Pythagorean identity
- A trigonometric equation derived from the Pythagorean theorem, relating squares of trig functions
- Unit circle
- A circle with radius 1 centered at the origin, where any point is
- Secant
- , the reciprocal of cosine
- Cosecant
- , the reciprocal of sine
- Cotangent
- , the reciprocal of tangent
Formula Card
Fundamental Identity
The sum of sine squared and cosine squared always equals 1
Tangent-Secant Identity
Derived by dividing the fundamental identity by $\cos^2\theta$
Cotangent-Cosecant Identity
Derived by dividing the fundamental identity by $\sin^2\theta$