1. An artist is designing triangular mirrors. Determine the number of different triangles that she can form using the given measurements. Then solve the triangles. Round to the nearest tenth.

a=4.2 cm b= 5.7 cm measure angle A= 39 degrees

2. What angle in the first quadrant could you reference to help you find the sine and cosine of each of the following angles?
a. 330 degrees
b. 120 degrees
c. 113 degrees
d. 203 degrees

3. Solve for x.
(3/x+1)+4/x=2

**I need help especially on this.4. Sketch a unit circle. In your circle, sketch in an angle that has:

a. A positive cosine and a negative sine.
b. A sine of -1.
c. A negative cosine and a negative sine.
d. A cosine of about -0.9 and a sine of about 0.4.
e. Could an angle have a sine equal to 0.9 and cosine equal to 0.8? Give an example angle or explain why not?

1.

I made a sketch and we have
sinB/5.7 = sin39°/4.2
sinB = .854
so B = 58.66° or B =121.34°

So we have the ambiguous case

if B = 58.66° , then C = 82.34°
find c using the sine law with those values
if B = 121.34° , then C = 19.66°
find c using the sine law

Two different triangles would be possible

2.
330° ----> 360-330 = 30°
120°-----> 180-120 = 60
113 =----> 180 - 113 = 67
etc

3. Did you mean
3/(x+1) or 3/x + 1 the way you typed it, ---< big difference!!!

4.
Unfortunately we cannot show graphs on this forum.
Here is a clip that has an interesting way to memorize the different situations.

http://www.youtube.com/watch?v=ao4EJzNWmK8

1. To determine the number of different triangles that can be formed using the given measurements, we can use the Sine Rule. The Sine Rule states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant.

In this case, we have side lengths a = 4.2 cm and b = 5.7 cm, and angle A = 39 degrees. Let's call the unknown side length c.

Using the Sine Rule, we have:

a/sin(A) = c/sin(C)

where C is the angle opposite to side c.

Rearranging the formula, we get:

sin(C) = (c * sin(A)) / a

Now, we can substitute the values we have:

sin(C) = (c * sin(39 degrees)) / 4.2 cm

To solve for c, we will need another piece of information, either another side length or angle measurement.

Once we have the lengths of all three sides, we can use the Cosine Rule to solve the triangles. The Cosine Rule states that in any triangle, the square of one side length is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides and the cosine of the included angle.

2. In the first quadrant, the reference angle is the acute angle formed between the x-axis and the terminal side of the given angle.

a. 330 degrees: The reference angle is 30 degrees (360 degrees - 330 degrees).
b. 120 degrees: The reference angle is 120 degrees.
c. 113 degrees: The reference angle is 113 degrees.
d. 203 degrees: The reference angle is 203 degrees.

To find the sine and cosine of each of these angles, we can use the values of the corresponding reference angles in the first quadrant. For example, to find the sine and cosine of 330 degrees, we can use the values of the sine and cosine of 30 degrees.

3. To solve the equation (3/x+1) + (4/x) = 2 for x, we can start by simplifying the equation.

First, let's find a common denominator for the fractions on the left side of the equation. The common denominator will be x(x+1).

Multiplying each term with the common denominator, we get:

3(x) + 4(x+1) = 2x(x+1)

Expanding and rearranging, we have:

3x + 4x + 4 = 2x^2 + 2x

Combining like terms, we get:

7x + 4 = 2x^2 + 2x

Next, let's move all terms to one side to set it equal to zero:

2x^2 + 2x - 7x - 4 = 0

Simplifying, we have:

2x^2 - 5x - 4 = 0

Now, we can either factor or use the quadratic formula to solve for x. Factoring is not possible in this case, so let's use the quadratic formula.

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, a = 2, b = -5, and c = -4.

Plugging in these values, we have:

x = (-(-5) ± sqrt((-5)^2 - 4(2)(-4))) / (2(2))

Simplifying further, we get:

x = (5 ± sqrt(25 + 32)) / 4

x = (5 ± sqrt(57)) / 4

Therefore, the solutions for x are (5 + sqrt(57)) / 4 and (5 - sqrt(57)) / 4.

4. To sketch a unit circle, draw a circle with a radius of 1 unit. The center of the circle represents the origin (0, 0) on a coordinate plane.

a. A positive cosine and a negative sine: Sketch an angle with a terminal side in the second quadrant, where the x-coordinate (cosine) is positive and the y-coordinate (sine) is negative.

b. A sine of -1: Sketch an angle with a terminal side in the third quadrant, where the y-coordinate (sine) is -1.

c. A negative cosine and a negative sine: Sketch an angle with a terminal side in the third quadrant, where both the x-coordinate (cosine) and the y-coordinate (sine) are negative.

d. A cosine of about -0.9 and a sine of about 0.4: Sketch an angle with a terminal side in the second quadrant, where the x-coordinate (cosine) is approximately -0.9 and the y-coordinate (sine) is approximately 0.4.

e. No, an angle cannot have a sine equal to 0.9 and cosine equal to 0.8. The sine and cosine values of an angle can never be greater than 1 or less than -1. Therefore, an angle with a sine of 0.9 and cosine of 0.8 does not exist on the unit circle.