Question

Explain why sin2(theta) + cos2(theta) = 1.

Answer 1

sin²θ + cos²θ = 1

Explanation:

We need to explain why sin²θ + cos²θ = 1

Consider a right angled triangle, we have base, perpendicular and hypotenuse.

Refer the figure

`sin\theta =\frac{\texttt{perpendicular}}{\texttt{hypotenuse}}\\\\cos\theta =\frac{\texttt{base}}{\texttt{hypotenuse}}`

Taking sin²θ + cos²θ

`sin^2\theta =\frac{\texttt{perpendicular}^2}{\texttt{hypotenuse}^2}\\\\cos^2\theta =\frac{\texttt{base}^2}{\texttt{hypotenuse}^2}\\\\sin^2\theta+cos^2\theta=\frac{\texttt{perpendicular}^2}{\texttt{hypotenuse}^2}+\frac{\texttt{base}^2}{\texttt{hypotenuse}^2}\\\\sin^2\theta+cos^2\theta=\frac{\texttt{perpendicular}^2+\texttt{base}^2}{\texttt{hypotenuse}^2}`

We know Pythagoras theorem

perpendicular² + base² = hypotenuse²

`sin^2\theta+cos^2\theta=\frac{\texttt{hypotenuse}^2}{\texttt{hypotenuse}^2}\\\\sin^2\theta+cos^2\theta=1`

Hence sin²θ + cos²θ = 1