Trigonometry Problems - Basics, Advanced & Examples

Solving Multl-Step Equations (continued)

Tell whether two angles can be as described. Justify your answers.
a. vertical and complementary
b. vertical and supplementary
c. complementary and supplementary

Draw a triangle that satisfies the set of conditions. Then classify the triangle.
a triangle with three acute angles and three congruent sides.

Veronica made a triangle by taking an 8 $\frac{1}{2}\times 11$ sheet of paper and putting a dot at the top. She then drew lines from the bottom corners to the dot and cut along the lines (see diagram).
What is the area of the triangle that Veronica cut out?

${\displaystyle \frac{1+\cos x}{\sin x}=\csc x+\cot x}$

The terminal side of an angle $\theta$ in standard position intersects the unit circle at ${\displaystyle (\frac{5}{13},\frac{12}{13})}$. What is $\sin (\theta )$?
Write your answer in simplified, rationalized form.

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 11 passengers per minute.

If the hypotenuse of a ${45}^{\circ }-{45}^{\circ }-{90}^{\circ }$ triangle has a length of ${\displaystyle \sqrt{2}}$, how long are the legs?

$\tan {28}^{\circ }=\cot ?$

In triangle ABC, $a=15$, $b=14$, $c=10$. Find ${\displaystyle m<B}$

Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is ${60}^{\circ }$, how far are you from the base of the plateau?

Prove the identity
$\frac{1}{2\csc 2x}={\cos }^{2}x\tan x$
Choose the sequence of steps below that verifies the identity
A) ${\cos }^{2}x\tan x={\cos }^{2}x\frac{\sin x}{\cos x}=\cos x\sin x=\frac{\sin 2x}{2}=\frac{1}{2\csc 2x}$
B) ${\cos }^{2}x\tan x={\cos }^{2}x\frac{\cos x}{\sin x}=\cos x\sin x=\frac{\sin 2x}{2}=\frac{1}{2\csc 2x}$
C) ${\cos }^{2}x\tan x={\cos }^{2}x\frac{\cos x}{\sin x}=\cos x\sin x=2\sin 2x=\frac{1}{2\csc 2x}$
D) ${\cos }^{2}x\tan x={\cos }^{2}x\frac{\sin x}{\cos x}=\cos x\sin x=2\sin 2x=\frac{1}{2\csc 2x}$

${\displaystyle \tan \left(x\right)\sin \left(x\right)+\sec \left(x\right){\cos }^2\left(x\right)=\sec \left(x\right)}$

(1,5 points) Find the intervals of concavity and the inflection points of the function ${\displaystyle f\left(x\right)=x{(2x-1)}^2}$

Two college friends are taking a weekend road trip. Friday they leave home and drive 87 miles north for a night of dinner and dancing in the city. The next morning they drive 116 miles east to spend a day at the beach. If they drive straight home from the beach the next day, how far do they have to travel on Sunday?

${\displaystyle \frac{\sin (a-b)}{\cos a}\cos b=\tan a-\tan b}$

If $A={40}^{\circ },B={60}^{\circ }$, and $a=20$, find b.

[Triangle] Find the value of x and y.

$\frac{\tan \alpha +\cot \alpha }{\sin \alpha -\cos \alpha }={\sec }^2\alpha +{\csc }^2\alpha$

Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
A(1,0,-1), B(5,-3,0), C(1,5,2)
Find $\mathrm{\angle }CAB,\mathrm{\angle }ABC,\mathrm{\angle }BCA$