Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
A triangle is shown with two sides that each measure 21. The angle between the two sides is 38 degrees. One of the other base angles is labeled x degrees.
What is the value of x?
(2 points)
To find the value of x, we can use the fact that the sum of the angles in a triangle is 180 degrees.
We know that one of the base angles is x degrees, so the other base angle can be found by subtracting the given angle (38 degrees) and x from 180 degrees.
Let's call the other base angle y degrees.
Thus, y = 180 - (38 + x)
Since the triangle is shown with two sides that each measure 21, we can use the Law of Cosines to find the value of x.
The Law of Cosines states that c^2 = a^2 + b^2 - 2abcos(C), where c is the side opposite angle C, and a and b are the other two sides of the triangle.
In this case, c is equal to 21 (one of the sides given in the problem), a is also equal to 21 (the other side given in the problem), and angle C is equal to x degrees (the angle we are trying to find).
Substituting these values into the Law of Cosines, we have:
21^2 = 21^2 + 21^2 - 2(21)(21)cos(x)
441 = 441 + 441 - 882cos(x)
441 = 882 - 882cos(x)
882cos(x) = 882 - 441
882cos(x) = 441
cos(x) = 441/882
cos(x) = 1/2
To find the value of x, we need to find the inverse cosine (also known as arccosine) of 1/2.
Using a calculator or a trigonometric table, we find that x is approximately 60 degrees.
Therefore, the value of x is 60 degrees.
To find the value of x, we can use the fact that the sum of the angles of a triangle is 180 degrees.
Step 1: We know that the angle between the two sides is 38 degrees.
Step 2: We also know that the sum of the measures of the two base angles of a triangle is equal to the angle between the two sides.
Step 3: So, the sum of the two base angles is 38 degrees.
Step 4: We can let one of the base angles be x degrees.
Step 5: Therefore, the other base angle would be (38 - x) degrees.
Step 6: Since the sum of the two base angles is 38 degrees, we can write the equation x + (38 - x) = 38.
Step 7: Simplifying the equation, we get 2x = 0.
Step 8: Dividing both sides of the equation by 2, we get x = 0.
Step 9: So, the value of x is 0 degrees.
To find the value of angle x in the triangle, we can use the fact that the sum of the angles in a triangle is always 180 degrees.
Given that one angle is 38 degrees and the other two sides measure 21 each, we can use the Law of Cosines to find the third angle.
The Law of Cosines states that for any triangle with sides a, b, and c, and the angle opposite to side c is C, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, let's assign a = 21, b = 21, and C = 38 degrees.
Substituting these values into the equation, we have:
21^2 = 21^2 + 21^2 - 2 * 21 * 21 * cos(38)
441 = 441 + 441 - 2 * 21 * 21 * cos(38)
Now, let's solve for cos(38):
cos(38) = (441 + 441 - 441) / (2 * 21 * 21)
cos(38) = 441 / (840)
cos(38) ≈ 0.525
Now, to find angle x, we can use the Law of Sines, which states that in any triangle with sides a, b, and c, and the angles opposite to those sides are A, B, and C respectively, the following equation holds:
sin(A) / a = sin(B) / b = sin(C) / c
In this case, let's assign A = 38 degrees, a = 21, and C = x degrees (the angle we want to find).
Substituting these values into the equation, we have:
sin(38) / 21 = sin(x) / 21
sin(38) = sin(x)
To find x, we can take the inverse sine (also called arcsine) of sin(38):
x = arcsin(sin(38))
x ≈ 38 degrees
Therefore, the value of angle x in the triangle is approximately 38 degrees.