a triangle has sides 8cm and 5cm and an angle of 90 degrees between them. Calculate the smallest angle of the triangle.
two ways to do this:
the hypotenuse is √89, so the other two angles are
arcsin(8/√89) and arcsin(5/√89)
Or, using the law of sines, if the right angle is C, we have
1/√89 = sinA/5 = sinB/8
The calculation gives the same results for angles A and B.
To calculate the smallest angle of the given triangle, we need to use the trigonometric functions: sine, cosine, and tangent.
The given triangle has sides 8cm and 5cm with a right angle (90 degrees) between them. This indicates that the triangle is a right triangle.
In a right triangle, the longest side is called the hypotenuse, and the other two sides are called the legs. The angle between the two legs is always 90 degrees.
To find the smallest angle of the triangle, we can focus on the angle opposite the smallest side, which is 5cm in this case.
We will use the sine function to calculate this angle. The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
sin(θ) = opposite/hypotenuse
In this case, the opposite side is the side with length 5cm, and the hypotenuse is the longest side with length 8cm.
sin(θ) = 5/8
To find the value of θ, we can use the inverse sine function (also known as arcsine or sin^-1) to calculate the angle.
θ = sin^-1(5/8)
Using a calculator or trigonometric tables, we can evaluate this expression and find that θ is approximately equal to 38.66 degrees.
Therefore, the smallest angle of the triangle is approximately 38.66 degrees.