A triangle has sides 8cm and 5cm and an angle of 90° between them. calculate the smallest Angle of the triangle.give the steps
You may wish to use the Pythagorean Theorem to find the length of the hypotenuse, then use SIN or COS to find each of the angles (primary trig ratios SOHCAHTOA).
the smallest angle θ is opposite the shortest side, and
tanθ = 5/8
So θ = tan-1 5/8
or, since the hypotenuse is √89, use the law of sines.
sinθ/5 = sin90°/√89
To find the smallest angle of a triangle, you can use the Law of Cosines or the Law of Sines. In this case, since you know two sides and the angle between them, it is more convenient to use the Law of Cosines.
The Law of Cosines states that in a triangle with sides a, b, and c, and the angle C opposite to side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we are given sides a = 8 cm, b = 5 cm, and the angle C = 90°. Let's substitute these values into the equation and solve for c^2:
c^2 = 8^2 + 5^2 - 2 * 8 * 5 * cos(90°)
c^2 = 64 + 25 - 80 * 0
c^2 = 89 - 0
c^2 = 89
Now, we have the value of c^2, which means c = √89.
By solving for the remaining angles using the Law of Sines or Cosines, we can determine the smallest angle of the triangle. However, since we are only interested in the smallest angle, we can use the fact that the smallest angle in a right triangle is always 90°.
Therefore, the smallest angle of the triangle is 90°.