A triangle has sides 8cm and 5cm and angle of 90(degree) between them calculate the smallest angle of the triangle
The smallest angle is opposite the shortest side, so
tanθ = 5/8
θ = 32°
I need the solution to the question please it is urgent
I need the full workings
Jan 8 2008
Mar 1 2021
To calculate the smallest angle of a triangle with two given sides and an angle between them, you can use the Law of Cosines. The Law of Cosines states that in any triangle with sides a, b, and c opposite to angles A, B, and C, respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, let's label the given sides as follows:
a = 8 cm
b = 5 cm
C = 90°
Now, we can substitute these values into the equation and solve for the value of c^2:
c^2 = 8^2 + 5^2 - 2 * 8 * 5 * cos(90°)
Simplifying this equation, we get:
c^2 = 64 + 25 - 80 * 0
Since cos(90°) equals 0, the equation further simplifies to:
c^2 = 64 + 25
c^2 = 89
To find the value of c, we take the square root of both sides:
c = √89
Therefore, the length of the third side (c) is approximately √89 cm.
Now, to find the smallest angle of the triangle, we can use the Law of Sines. This law states that in any triangle with sides a, b, and c, and angles A, B, and C (opposite to sides a, b, and c, respectively), the following equation holds:
a/sin(A) = b/sin(B) = c/sin(C)
Since we know the lengths of sides a, b, and c, and we need to find the smallest angle, we can rearrange the equation as follows:
sin(A) = a / c
Substituting the known values, we have:
sin(A) = 8 / √89
To find the value of sin(A), we can use a scientific calculator or an online trigonometric calculator. By taking the inverse sine (sin^-1) of both sides, we can find the value of A, which represents the smallest angle of the triangle.
Once you calculate sin^-1(8 / √89), you will obtain the value of A, which is the smallest angle of the triangle.