Answer:
your worksheet shows the answers
Explanation:
I can give you a clue as to what to do, but 6 problems is more than I'm willing to work for you.
__
Given two sides and the angle between them (angle has a different letter than either side), the Law of Cosines is used to find the third side of the triangle. It is often written as ...
c² = a² +b² -2ab·cos(C)
The same equation is applicable for any permutation of a, b, c and the corresponding permutation of A, B, C.
1.
The formula above can be used directly.
c² = 6² +7² -2(6)(7)·cos(20°) = 36 +49 -84cos(20°) ≈ 6.06582
c ≈ √6.06582 ≈ 2.46
__
3.
The version of the formula applicable to this problem is ...
b² = a² +c² -2ac·cos(B)
__
5.
The version of the formula applicable to this problem is ...
a² = b² +c² -2bc·cos(A)
__
7.
The same formula is used for finding an angle when three sides of the triangle are given. For this problem, we're concerned with angle B, so we use the same version of the formula we used in problem 3. Here, we need to solve for the cosine of the angle.
b² = a² +c² -2ac·cos(B)
2ac·cos(B) = a² +c² -b² . . . . . . . add 2ac·cos(B) - b² to both sides
cos(B) = (a² +c² -b²)/(2ac) . . . . divide by the coefficient of the cosine
B = arccos((a² +c² -b²)/(2ac)) . . . . use the inverse cosine function to find B
For the given side lengths, the angle is ...
B = arccos((8² +12² -10²)/(2(8)(12)) = arccos(108/192)
There is no need to reduce the fraction (to 9/16). Your calculator must be set to degrees mode.
B ≈ 55.7711° ≈ 55.8°
__
9.
The smallest angle is opposite the shortest side, so will be angle A. The version of the formula you need for this problem is ...
A = arccos((b² +c² -a²)/(2bc))
__
11.
The largest angle is opposite the longest side, so will be angle C. The version of the formula you need for this problem is ...
C = arccos((a² +b² -c²)/(2ab)) = arccos((1.6² +0.9² -1.8²)/(2(1.6)(0.9))
= arccos(0.13/2.88) ≈ 87.413°
angle C ≈ 87.4°
_____
Additional comments
It will be worth your while to notice the pattern in these formulas. The formula for the missing side has that side variable on the left, and the other two side variables are used on the right side. The angle opposite the unknown side is the argument of the cosine function.
When the formula is rearranged to find the angle, the square being subtracted in the numerator is the square of the side opposite the angle. The other variables are the other two sides.
__
You can always check your work using any of numerous triangle calculators available online, as apps, or perhaps even on your graphing calculator.
__
The name of the inverse cosine function is often written as cos⁻¹(x). It is generally found on a calculator as a "second" or "alternate" function of the cos(x) key.