In triangle PEA it is known that PE = 15 m, EA = 10 m, and P = 15 degrees. How long is PA?
To find the length of PA, we can use the law of sines.
The law of sines states that in any triangle,
sin(A)/a = sin(B)/b = sin(C)/c
where A, B, and C are the angles and a, b, and c are the side lengths opposite to those angles.
In this case, we know that P = 15 degrees, EA = 10 m, and PE = 15 m. Let's label PA as x.
Using the law of sines, we can write:
sin(P)/PE = sin(E)/EA
sin(15)/15 = sin(E)/10
Cross multiplying, we get:
10 * sin(15) = 15 * sin(E)
sin(E) = (10/15) * sin(15)
sin(E) ≈ 0.555 * 0.259
sin(E) ≈ 0.1434
Now we can find the value of angle E using the arcsin function:
E ≈ arcsin(0.1434)
E ≈ 8.18 degrees
Since the sum of the angles in a triangle is always 180 degrees, we can find angle A:
A = 180 - P - E
A = 180 - 15 - 8.18
A ≈ 156.82 degrees
Now we can find the length of PA using the law of sines:
sin(P)/PA = sin(A)/EA
sin(15)/PA = sin(156.82)/10
Cross multiplying, we get:
10 * sin(15) = PA * sin(156.82)
sin(15)/sin(156.82) = PA/10
PA ≈ (sin(15)/sin(156.82)) * 10
PA ≈ (0.259/0.987) * 10
PA ≈ 2.63 * 10
PA ≈ 26.3 m
Therefore, the length of PA is approximately 26.3 meters.
wrong
Apologies for the incorrect earlier response. Let's try solving it again.
In triangle PEA, P = 15 degrees, PE = 15 m, and EA = 10 m. We want to find the length of PA.
Since we have the lengths of two sides and the measure of one angle, we can use the Law of Cosines to solve for the third side. The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the side opposite angle C, and a and b are the lengths of the other two sides.
In this case, we are looking to find the length of side PA (c). So we can rewrite the equation as:
PA^2 = PE^2 + EA^2 - 2 * PE * EA * cos(P)
Substituting the known values into this equation, we get:
PA^2 = 15^2 + 10^2 - 2 * 15 * 10 * cos(15)
PA^2 = 225 + 100 - 2 * 150 * cos(15)
Now we can calculate this expression:
PA^2 = 360.975
Taking the square root to isolate PA, we get:
PA ≈ 19.01 m
Therefore, the approximate length of PA is 19.01 meters.
Opposite of angle P is side EA so equation is: PE^2 = PA^2 + EA^2 - 2 * PA * EA * cos(P)
You're correct, my previous response had an error in the equation. Let's try solving it again using the correct equation.
In triangle PEA, P = 15 degrees, PE = 15 m, and EA = 10 m. We want to find the length of PA.
Using the Law of Cosines, we have the equation:
PE^2 = PA^2 + EA^2 - 2 * PA * EA * cos(P)
Substituting the given values, we get:
15^2 = PA^2 + 10^2 - 2 * PA * 10 * cos(15)
225 = PA^2 + 100 - 20 * PA * cos(15)
Rearranging the equation:
PA^2 - 20 * PA * cos(15) + 125 = 0
This is a quadratic equation in terms of PA. We can solve it using the quadratic formula:
PA = (-(-20 * cos(15)) ± √((-20 * cos(15))^2 - 4 * 1 * 125))/(2 * 1)
PA = (20 * cos(15) ± √(400 * cos(15)^2 - 500))/2
PA = (20 * cos(15) ± √(400 * (0.966)^2 - 500))/2
PA = (20 * cos(15) ± √(400 * 0.933156 - 500))/2
PA ≈ (20 * cos(15) ± √(373.2624 - 500))/2
PA ≈ (20 * cos(15) ± √(-126.7376))/2
Since we cannot take the square root of a negative number in this context, this quadratic equation does not have real solutions.
Therefore, there is no valid value for the length of PA that satisfies the given conditions.
To find the length of side PA in triangle PEA, you can use the law of sines. The law of sines states that the ratio of the length of a side to the sine of its opposite angle is constant. The formula is given as:
a/sinA = b/sinB = c/sinC
In our case, we know the lengths of side PE and EA, and the measure of angle P. We want to find the length of side PA, which is opposite to angle A.
First, let's find the measure of angle E using the fact that the sum of the angles in a triangle is 180 degrees:
P + E + A = 180 degrees
15 + E + 10 = 180
E = 180 - 25
E = 155 degrees
Now we can use the law of sines to find the length of side PA:
PA/sinP = EA/sinE
Substituting the known values:
PA/sin(15) = 10/sin(155)
Using a scientific calculator, we can find that sin(155) ≈ 0.5736 and sin(15) ≈ 0.2588.
PA/0.2588 = 10/0.5736
Simplifying the equation:
PA ≈ (10 * 0.2588) / 0.5736
PA ≈ 4.985 / 0.5736
PA ≈ 8.692 m
Therefore, the length of side PA is approximately 8.692 meters.
To find the length of side PA in triangle PEA, we can use the law of cosines. The law of cosines states that the square of one side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides multiplied by the cosine of the included angle.
Let's label the length of side PA as x. The law of cosines can be written as:
x^2 = PE^2 + EA^2 - 2 * PE * EA * cos(P)
Substituting the given values:
x^2 = 15^2 + 10^2 - 2 * 15 * 10 * cos(15 degrees)
x^2 = 225 + 100 - 2 * 15 * 10 * cos(15 degrees)
Now, we can use a calculator to find the cosine of 15 degrees:
cos(15 degrees) ≈ 0.9659
Substituting this value:
x^2 = 225 + 100 - 2 * 15 * 10 * 0.9659
x^2 = 325 - 2 * 15 * 10 * 0.9659
x^2 = 325 - 287.85
x^2 = 37.15
To find the length of PA, we take the square root of both sides:
x ≈ √37.15
Using a calculator, we find:
x ≈ 6.1
Therefore, the approximate length of side PA is 6.1 meters.