Daily math problems for August 7, 2026 — Algebra, Calculus, Trigonometry & more

A ring: abelian group under $+$, associative $\cdot$, and distributivity of $\cdot$ over $+$. (Some definitions also require a multiplicative identity.)

$I_0 = \int_0^1 1 \, dx = 1$.

For polynomials, substitute directly: $4 + 6 - 1 = 9$.

$\bullet$ Recognise a binomial square:

$$\sin^4 7°+2\sin^2 7°\cos^2 7°+\cos^4 7°=(\sin^2 7°+\cos^2 7°)^2$$

$\bullet$ Pythagoras kills the angle:

$$(\sin^2 7°+\cos^2 7°)^2=1$$

$\bullet$ Evaluate the limit:

$$\lim_{x\to 0}\dfrac{\sin 5x}{x}=5$$

$\bullet$ Decode the log:

$$\log_{\sqrt 7}49=\dfrac{2}{1/2}=4$$

$\bullet$ Combine:

$$1+5-4=2$$

Lagrange: in a finite group, the order of any subgroup (and hence the order of any element, which is the order of the cyclic subgroup it generates) divides $|G|$.

$5$ is prime, so $\mathbb{Z}_5$ is a field. (Every non-zero element has a multiplicative inverse.)

$\bullet$ Evaluate every entry:

$$\cos 60°=\sin 30°=\tfrac{1}{2},\ \tan 45°=1,\ \log_2 4=2$$

$\bullet$ The matrix becomes:

$$A=\begin{pmatrix}1/2&1\\2&1/2\end{pmatrix}$$

$\bullet$ Compute the determinant:

$$\det A=\tfrac{1}{4}-2=-\tfrac{7}{4}$$

$\bullet$ Apply $\det(A^{-1})=1/\det A$:

$$\det(A^{-1})=-\tfrac{4}{7}$$

After both moves, the value is $0.8 \cdot 1.2 = 0.96$ — a $4\%$ net loss.

$\dfrac{1}{n}\sum (x_i + 5) = \dfrac{1}{n}\sum x_i + 5 = \bar{x} + 5$.

Folosind schimbarea de bază, $\log_k (k+1) = \dfrac{\ln(k+1)}{\ln k}$. Produsul telescopează: $\dfrac{\ln 3}{\ln 2} \cdot \dfrac{\ln 4}{\ln 3} \cdots \dfrac{\ln 256}{\ln 255} = \dfrac{\ln 256}{\ln 2} = \log_2 256 = 8$.