Basic Trigonometric Identity formula:
story, analogy and memory trick
The Basic Trigonometric Identity formula is sin²θ + cos²θ = 1. For any angle θ, the square of the sine plus the square of the cosine is always equal to 1.
The Basic Trigonometric Identity formula
- θany angle
- sin θopposite side ÷ hypotenuse
- cos θadjacent side ÷ hypotenuse
The story
Picture a point moving around a circle of radius 1. Its height is sin θ and its horizontal distance from the centre is cos θ. Those two distances and the radius 1 form a right triangle, so Pythagoras gives sin²θ + cos²θ = 1, wherever the point stops.
Real-life analogy
Think of two friends sharing one pizza. When one takes more, the other has less, but their squared shares always make exactly one whole.
Memory trick
Say 'sin square plus cos square is one' like a rhyme. Divide by cos²θ to get 1 + tan²θ = sec²θ, and by sin²θ to get 1 + cot²θ = cosec²θ.
Exam tip
It holds for every angle, so use it to swap sin²θ for 1 − cos²θ and the other way round. The square belongs to sin and cos, not to θ.
Questions about Basic Trigonometric Identity
What is the Basic Trigonometric Identity formula?
How do I remember the Basic Trigonometric Identity formula?
What should I watch out for in exams with Basic Trigonometric Identity?
More Maths formulas
Quadratic
x = (−b ± √(b² − 4ac)) ÷ 2a
It gives the roots of any quadratic equation ax² + bx + c = 0 using only its three coefficients.
Read the storyPythagoras Theorem
a² + b² = c²
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Read the storyCompound Interest
A = P × (1 + r/100)ⁿ
With compound interest, each year's interest is added to the principal, so the amount after n years is P times (1 + r/100) raised to the power n.
Read the storyNeed a story for a different formula?
Type any formula or topic into the free ThunderStudy AI tool and get a story, a real-life analogy or a memory trick.