A triangle has 10cm and 6cm and an angle of 90degree between them, calculate the smallest angle in the triangle
Since 6 is the smallest side, the smallest angle will be opposite it.
the angle will be x, such that tan(x) = 6/10
To calculate the smallest angle in a triangle with side lengths 10 cm and 6 cm, and an angle of 90 degrees between them, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the side lengths are 10 cm and 6 cm, and the angle between them is 90 degrees.
Using the Pythagorean theorem, we can find the length of the hypotenuse (the longest side) as follows:
c^2 = a^2 + b^2,
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.
Plugging in the given values, we have:
c^2 = 10^2 + 6^2,
c^2 = 100 + 36,
c^2 = 136.
Taking the square root of both sides to solve for c, we have:
c = √136,
c ≈ 11.66 cm.
Now that we know the lengths of all three sides, we can find the angles of the triangle using the Law of Cosines. The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C),
where C is the angle opposite side c.
Plugging in the values, we have:
11.66^2 = 10^2 + 6^2 - 2(10)(6) * cos(C),
136 = 100 + 36 - 120 * cos(C),
136 = 136 - 120 * cos(C).
Simplifying the equation, we have:
0 = -120 * cos(C).
In order for the equation to hold true, cos(C) must be equal to 0. Therefore, the angle C (the smallest angle in the triangle) is equal to 90 degrees.
Hence, the smallest angle in the triangle is 90 degrees.
To calculate the smallest angle in a triangle, we need to use the fact that the sum of all angles in a triangle is always 180 degrees.
In this case, we have a right triangle with one angle measuring 90 degrees. Let's call the other two angles A and B.
Since the sum of all angles in a triangle is 180 degrees, we can write the equation: A + B + 90 = 180.
To find the smallest angle, we need to solve for A or B. Let's solve for A.
Subtracting 90 from both sides of the equation, we get: A + B = 90.
Now we can substitute the lengths of the sides of the triangle into a trigonometric equation to solve for the angle A.
In a right triangle, the sides are related by the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the two legs of the triangle, and c is the hypotenuse.
In this case, the two legs are 10 cm and 6 cm, and the hypotenuse is c.
Using the Pythagorean theorem, we can solve for c:
10^2 + 6^2 = c^2,
100 + 36 = c^2,
136 = c^2.
Taking the square root of both sides, we find that the hypotenuse c is approximately 11.66 cm.
Now we can use trigonometric ratios to find the angle A.
sin(A) = opposite / hypotenuse,
sin(A) = 6 / 11.66.
Taking the inverse sine (arcsin) of both sides, we get:
A ≈ arcsin(6 / 11.66).
Using a calculator, we find that A ≈ 31.32 degrees.
Therefore, the smallest angle in the triangle is approximately 31.32 degrees.