1)Find the exact value of cot(-315)
A)1
B)square root of 2
C)square root of 2 over 2
D)2
I chose C
2)Find the exact value of sin(-pi/6)
A)-1/2
B)-square root of 3/2
C)-square root of 2/2
D)square root of 2/2
I chose A
3)In triangle ABC, C=30, c=22, and b=42. Find B.
A)73 degrees
B)107 degrees
C)77 degrees
D)15 degrees
I chose A
4)Find the area of triangle ABC, if A is 55 degrees, b=8 meters, and c=14 meters.
A)91.7 sq meters
B)32.1 sq meters
C)45.9 sq meters
D)56.0 sq meters
I chose C
5)Which triangle has two solutions?
A)A=130 degrees, a=19, b=11
B)A=45 degrees, a=4square root of 2, b=8
C)A=32 degrees, a=16, b=21
D)A=90 degrees, a=25, b=15
I chose C
all ok except #1
cot (-315)
= cot 45
= 1/tan 45
= 1/1
= 1
In abc,c=60 degrees,a = 12,and b=5 . Find c.
1) To find the exact value of cot(-315), we first need to determine the reference angle. Since -315 degrees is equivalent to 45 degrees in the clockwise direction, the reference angle is 45 degrees.
Next, we can use the definition of cotangent. The cotangent of an angle is equal to the cosine of that angle divided by the sine of that angle.
Since the reference angle is 45 degrees, we can find the exact values of cosine and sine for this angle.
The cosine of 45 degrees is equal to the square root of 2 over 2 (option C) and the sine of 45 degrees is also equal to the square root of 2 over 2 (option C).
Therefore, cot(-315) is equal to the cosine of 45 degrees divided by the sine of 45 degrees, which gives us (sqrt(2)/2) / (sqrt(2)/2) = 1.
So the correct answer is option A) 1.
2) To find the exact value of sin(-pi/6), we need to determine the reference angle.
The angle -pi/6 is in the fourth quadrant, so we can find the reference angle by taking the positive version of pi/6, which is pi/6 itself.
The sine of pi/6 is equal to 1/2 (option D).
However, since we are in the fourth quadrant, the sine value is negative. Therefore, sin(-pi/6) is equal to -1/2.
So the correct answer is option A) -1/2.
3) To find angle B in triangle ABC, we can use the Law of Cosines.
The Law of Cosines states that c^2 = a^2 + b^2 - 2ab*cos(C).
In this case, we know that c = 22, a = 42, and C = 30 degrees.
By plugging in these values into the formula, we have:
22^2 = 42^2 + b^2 - 2 * 42 * b * cos(30).
Simplifying further, we have:
484 = 1764 + b^2 - 84b * sqrt(3).
Rearranging the equation, we get:
b^2 - 84b * sqrt(3) + 1280 = 0.
Solving this quadratic equation, we find that b ≈ 33.29.
Now, to find angle B, we can use the Law of Sines. The Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C).
Plugging in the values we have, we have:
42 / sin(30) = 33.29 / sin(B).
Rearranging the equation, we can solve for sin(B):
sin(B) = (33.29 * sin(30)) / 42.
Therefore, sin(B) ≈ 0.4115.
To find angle B, we can use the inverse sine function (sin^-1) on a calculator, which gives us approximately 24.98 degrees.
So the correct answer is option A) 73 degrees.
4) To find the area of triangle ABC with angle A = 55 degrees, side b = 8 meters, and side c = 14 meters, we can use the formula for the area of a triangle:
Area = (1/2) * b * c * sin(A).
Plugging in the values we have, we have:
Area = (1/2) * 8 * 14 * sin(55).
Calculating the sine of 55 degrees, we get:
Area = (1/2) * 8 * 14 * 0.8192.
Area ≈ 45.9 square meters.
So the correct answer is option C) 45.9 square meters.
5) To determine which triangle has two solutions, we can use the Law of Sines and look for any ambiguous case where one angle has multiple possible values.
In the ambiguous case, the sine function can have two angles that produce the same sine value. This occurs when the length of the opposite side (a) is greater than or equal to the length of the hypotenuse (c).
Examining the given options, none of them satisfy this condition except option C) A = 32 degrees, a = 16, and b = 21.
In this case, since a = 16 is greater than c = 21, we have an ambiguous case.
So the correct answer is option C) A = 32 degrees, a = 16, and b = 21.