A triangle has sides 8cm and 5cm and an angle of 90° between them. Calculate the smallest angle of the triangle.
To calculate the smallest angle of the triangle, we need to apply the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, we have a right-angled triangle with sides 8 cm and 5 cm. The hypotenuse is the longest side, which we can find using the Pythagorean theorem.
Using the formula, we have:
c^2 = a^2 + b^2
Where c represents the length of the hypotenuse (unknown), and a and b represent the lengths of the other two sides.
Plugging in the values, we get:
c^2 = 8^2 + 5^2
c^2 = 64 + 25
c^2 = 89
To find the length of the hypotenuse, we take the square root of both sides:
c = √(89)
c ≈ 9.43 cm
Now that we know the lengths of all three sides, we can determine the angles using trigonometric functions.
To find the smallest angle of the triangle, we can use the inverse sine function (sin⁻¹) because we know the lengths of the opposite and hypotenuse sides.
Let's calculate it:
sin(small angle) = opposite / hypotenuse
sin(small angle) = 5 / 9.43
Using a calculator or trigonometric tables, we can input the value of 5 / 9.43 and find the inverse sine:
small angle ≈ 33.8°
Therefore, the smallest angle of the triangle is approximately 33.8°.