comprehension check for derivatives of trigonometric functions: a) true or false: if fprime(theta) = -sin(theta), then f(theta) = cos(theta). b) true

Given a right triangle. One cathetus is 100cm long. Find the length of the other cathetus, if the angle opposite to it is ${\displaystyle {71.6}^{\circ }}$. Round your answer to an integer.

Given a right triangle. Cathetuses are 24cm and 7cm long. Find the length of the hypothenuse.

Consider the non-triangle shown below that has side lenghts of 1.6, 1.751 and 2.6 cm and interior angle measures of 0.72, alpha and beta degrees.
What is value of alpha and beta

Given a right triangle, where one cathetus is ${\displaystyle \bar{A}=51}$ m and the second cathetus is ${\displaystyle \bar{B}=39}$ m. Find the hypothenuse ${\displaystyle \left(\bar{C}\right)}$ and the angle opposite to ${\displaystyle \bar{B}\left(\mathrm{\angle }a\right)}$

Prove that ${\displaystyle |a-b|\ge \left|a\right|-\left|b\right|}$.

Consider the non-triangle below. Suppose that ${\displaystyle \mathrm{\angle }ACB={103}^{\circ }}$ and ${\displaystyle m\mathrm{\angle }BAC={44}^{\circ }}$, and that $y=50.5cm$. What of the value of X?

In triangle ABC, angle B is five times as large as angle A. The measure of angle C is 2° less than that of angle A. Find the measures of the angles. (Hint: The sum of the angle measures is 180°.)