How can you derive the Law of Sines for angles B and C?

To solve a triangle, you find all its angle measures and all its side lengths. It is important that you know when you can use the Law of Sines and when you can use the Law of Cosines to solve triangles. Suppose you know the measures of the three angles of a triangle. Can you use the Law of Sines and/ or the Law of Cosines to solve the triangle? Explain.

Derive the Law of Sines.

If one side and two angles of a triangle are given, which law can be used to solve the triangle?

(a) Law of Sines

(b) Law of Cosines

(c) Either a or b

(d) The triangle cannot be solved.

What elements in "The Masque of the Red Death" illustrate what is meant by King's metaphor of a closed, forbidden door on page 446?

The ambiguous case only causes a problem when the given angle is acute, not when an obtuse angle is given. Explain why.

The function

$$f(x)=ln(e^x)$$

is one of our twelve basic functions written in another form. Graph the function and identify which basic function it is.

The sum of the first n terms of an arithmetic sequence is

$$S _ { n } = 3 n ^ { 2 } - 2 n$$

. Find the nth term,

$$\mathcal { U } _ { n }$$

.

Solving by determine graphically the magnitude and direction of the resultant of the two forces shown using

(a) the parallelogram law,

(b) the triangle rule by trigonometry

Solve the equation.

$$x^2+x+11=5x-8$$

Write each number in scientific notation. -0.00000000405

Solve the equation. Give the exact solution and an approximate solution to four decimal places, when appropriate.

$$e ^ { 0.08 t } = 4$$

Complete each statement by writing the correct word or words. At equilibrium, both the forward and the reverse reactions continue, but the ______ of the reactants stay the same.

Use either Gaussian elimination or Gauss–Jordan elimination to solve the given system or show that no solution exists.

$$\begin{aligned} x_{1}+2 x_{2}-4 x_{3} &=9 \\ 5 x_{1}-x_{2}+2 x_{3} &=1 \end{aligned}$$