Trigonometry Problems - Basics, Advanced & Examples

If $A=40°,B=60°$, and $a=20$, find $b$.

Given a right triangle. The cathetuses are: 86m and 37m. Find the angle opposite to 86m cathetus.

Given a right triangle, where the cathetuses are 1 cm and $\sqrt{3}$ cm. Find the hypothenuse's length.

${\displaystyle \tan \left(t\right)=\frac{1}{4}}$
Find ${\displaystyle \sin \left(t\right)}$, if terminal point of t is in Quadrant III.

Proof trigonometric identities.
${\displaystyle \frac{1-\sin (-x)}{\cos x+\cos (-x)\sin x}=\sec x}$

Establish identity
${\displaystyle \frac{{1-\sin }^2\theta }{1-\cos \theta }=-\cos \theta }$

$\tan 28°=\cot$___

Verify the identity.
${\displaystyle {\cot }^2\theta -{\cos }^2\theta ={\cot }^2\theta {\cos }^{\theta }}$

${\displaystyle \sin (2{\cos }^{-1}\cdot \left(\frac{\sqrt{2}}{2}\right)}$

${\displaystyle C\left(x\right)=5,000+20x-\frac{1}{4}{x}^2}$
is a cost function.
Calculate the marginal average cost when 20

(a), use your result to estimate the total cost when $x=21$

(b) and calculate the percentage error between this estimate and the actual cost when $x=21$

Find the inverse the function and the domain and the range of f and ${\displaystyle f^{-1}}$, if the function is ${\displaystyle f\left(x\right)=\frac{2x}{x+5}}$ and it's one-to-one.

Find the differential of the following functions:
1. ${\displaystyle y=\tan \left(\sqrt{3t}\right)}$
2. ${\displaystyle y=5-\frac{v^2}{5}+v^2}$

Verify the identity
${\displaystyle \tan \times {\csc }^{2}x–\tan x=\cot }$

if $x=2,f(x)=1$
if $x=3,f(x)=4$
if $x=5,f(x)=-2$
if $x=8,f(x)=3$
if $x=13,f(x)=6$
f is twice variable for all real numbers.
1. Find $f^{\prime }(4)$
2. Approximate ${\displaystyle {\underset{2}{\int }}^{13}{f}^{\prime }\left(x\right)dx}$
2. Find the value of ${\displaystyle {\underset{2}{\int }}^8(3-f^{\prime }\left(x\right))dx}$

Given a right triangle, where one cathetus is ${\displaystyle \bar{A}=11}$ m, the hypothenuse is ${\displaystyle \bar{C}=15}$ m, and ${\displaystyle \mathrm{\angle }c={90}^{\circ }}$ . Find all missing sides ${\displaystyle \left(\bar{B}\right)}$ and angles ${\displaystyle \left(\mathrm{\angle }a\text{and}\mathrm{\angle }b\right)}$.

Verify the identity.
${\displaystyle \sin 2x\cos x-\cos 2x\sin x=\sin x}$

Establish identity
${\displaystyle 1-\frac{{\cos }^2\theta }{1+\sin \theta }=\sin \theta }$

If ${\displaystyle y=3\sqrt{x}+\frac{1}{\sqrt{x}}}$, find the differential dy.

A ring lies in the xy plane, centered at the origin. It has a radius of R and a uniformly distributed total charge Q. Due to the ring on the z-axis, as a function of z, what is the potential V(z)?

Proof trigonometry identities.
${\displaystyle \cot \left(x\right)+\tan \left(x\right)+\sec \left(x\right)\csc \left(x\right)=2\sec \left(x\right)\csc \left(x\right)}$