In triangle ABC, angle C is a right angle. Find the remaining sides and angles. Round your answer to the nearest tenth.
angle B = 52 degrees, b = 10
Since A+B+C = 180°, A = 38°
If you haven't had trig yet, it will be tough to calculate the other sides.
a = 10 tan 38° = 7.8
c^2 = a^2 + b^2 = 100 + 61 = 161
c = 12.7
To find the remaining sides and angles in triangle ABC, we can use the trigonometric ratios.
1. Let's start by finding angle A. Since angle C is a right angle, the sum of the angles in a triangle is 180 degrees. Therefore, angle A = 180 - (90 + 52) = 180 - 142 = 38 degrees.
2. Now, to find side a, we can use the sine ratio: sin(A) = opposite / hypotenuse.
Let's call side a as the opposite side to angle A, and side b as the adjacent side to angle A.
We have sin(A) = a / 10.
Rearranging the equation, a = 10 * sin(A).
Substitute the value of angle A, a = 10 * sin(38).
Using a calculator, we get a ≈ 6.111.
Therefore, side a ≈ 6.1 units.
3. Lastly, to find side c, we can use the Pythagorean theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
c^2 = a^2 + b^2.
Substitute the known values, c^2 = 6.1^2 + 10^2.
Simplifying the equation, c^2 = 37.21 + 100.
c^2 ≈ 137.21.
Taking the square root of both sides, c ≈ √137.21.
Using a calculator, we get c ≈ 11.7.
Therefore, side c ≈ 11.7 units.
To summarize:
Angle A ≈ 38 degrees,
Side a ≈ 6.1 units,
Side c ≈ 11.7 units.
To solve this problem, we can use the properties of right-angled triangles and the trigonometric ratios.
Given that angle C is a right angle, we know that it measures 90 degrees.
To find the remaining side and angles, we can use the trigonometric ratio called the sine function.
Using the sine function:
sin(B) = Opposite / Hypotenuse
Given that angle B is 52 degrees, we can substitute the values we know:
sin(52) = b / c
Solving for c (the hypotenuse):
c = b / sin(52)
Substituting the given value of b = 10 and using a calculator to find the sine of 52 degrees, we get:
c ≈ 10 / sin(52) ≈ 13.1
So, the hypotenuse, side c, is approximately 13.1 units.
To find the remaining side a, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
Substituting the given values:
a^2 + 10^2 = 13.1^2
Solving for a:
a^2 = 13.1^2 - 10^2
a ≈ sqrt(13.1^2 - 10^2) ≈ 6.8
So, the remaining side a is approximately 6.8 units.
To find angle A, we can use the trigonometric ratio called the sine function again:
sin(A) = Opposite / Hypotenuse
Using the known values:
sin(A) = a / c
Substituting the values we found:
sin(A) = 6.8 / 13.1
Using a calculator to find the arcsine of this value, we get:
A ≈ arcsin(6.8 / 13.1) ≈ 28.9 degrees
So, angle A is approximately 28.9 degrees.
To summarize, we have found the remaining sides and angles of triangle ABC:
Side a ≈ 6.8 units
Side b = 10 units
Side c ≈ 13.1 units
Angle A ≈ 28.9 degrees
Angle B = 52 degrees
Angle C = 90 degrees (since it is a right angle)