fill in the blank to complete the trigonometric formula.
A standard completion uses the Pythagorean identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1. \] The missing term in \( \sin^2(\theta) + \_\_\_\_ = 1 \) is \( \cos^2(\theta) \).
A common convention assumes \( \theta \) is a real angle measured in radians or degrees. The identity holds for all real \( \theta \) because it comes from the geometry of the unit circle.
Identity completion
The blank in \[ \sin^2(\theta) + \_\_\_\_ = 1 \] is the term that pairs with \( \sin^2(\theta) \) in the Pythagorean identity. The completed formula is \[ \sin^2(\theta) + \cos^2(\theta) = 1. \]
Unit circle justification
On the unit circle, a point at angle \( \theta \) has coordinates \( (\cos(\theta), \sin(\theta)) \). The radius equals 1, so the distance from the origin satisfies \[ \cos^2(\theta) + \sin^2(\theta) = 1^2 = 1. \] Reordering the sum gives the same identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1. \]
Algebraic rearrangements used in fill-in formats
Equivalent forms arise by isolating a term. These are common in “fill in the blank” prompts involving trigonometric formulas:
| Form | Completed expression | Meaning |
|---|---|---|
| \( \sin^2(\theta) + \_\_\_\_ = 1 \) | \( \sin^2(\theta) + \cos^2(\theta) = 1 \) | Complementary squared terms sum to 1 |
| \( 1 - \sin^2(\theta) = \_\_\_\_ \) | \( 1 - \sin^2(\theta) = \cos^2(\theta) \) | Cosine squared as the remainder |
| \( 1 - \cos^2(\theta) = \_\_\_\_ \) | \( 1 - \cos^2(\theta) = \sin^2(\theta) \) | Sine squared as the remainder |
Common pitfalls
- Square placement: \( \sin^2(\theta) \) means \( (\sin(\theta))^2 \), not \( \sin(\theta^2) \).
- Sign errors: the Pythagorean identity uses a plus sign, not a minus sign.
- Angle consistency: both terms use the same angle \( \theta \) in the identity.