In a right triangle ABC, C = 90°, B = 38° 35' and a = 6.434 mi. Solve for all the missing parts using the given information. (Round each answer to three decimal places for sides and to two decimal places for angle.)
You draw a triangle and label all of your given information.
so you subtract 90 from 38degree35' to get angle A.
then you use SOH CAH TOA
tan 38degree35'= b/6.434mi to solve for side b.
then use cos38degree35' = 6.434mi/c to solve for side c.
i don't know and don't care.
To solve for the missing parts of the right triangle ABC, we can use the trigonometric functions sine, cosine, and tangent.
Given:
C = 90°
B = 38° 35'
a = 6.434 mi
To find side c:
In a right triangle, side c is the hypotenuse, which is opposite the right angle. We can use the sine function to solve for side c:
sin(C) = c / a
Since C = 90°, the sine of 90° is equal to 1. Therefore, we have:
1 = c / 6.434
To find c, we can rearrange the equation and solve for it:
c = 1 * 6.434
c ≈ 6.434 mi
So, side c is approximately 6.434 miles.
To find side b:
In a right triangle, side b is the side adjacent to angle B. We can use the cosine function to solve for side b:
cos(B) = b / a
To use the cosine function, we need to convert the angle B from degrees and minutes to decimal degrees:
38° 35' = 38 + (35/60) = 38.5833°
Now we can solve for side b:
cos(38.5833°) = b / 6.434
To find b, we can rearrange the equation and solve for it:
b = cos(38.5833°) * 6.434
Using a scientific calculator or a trigonometric table, we can find the cosine of 38.5833°:
cos(38.5833°) ≈ 0.788
Now we can solve for b:
b ≈ 0.788 * 6.434
b ≈ 5.066 mi
So, side b is approximately 5.066 miles.
To find angle A:
In a right triangle, the sum of the angles is always 180°. Therefore, we can find angle A by subtracting angles B and C from 180°:
A = 180° - B - C
Substituting the given values:
A = 180° - 38° 35' - 90°
First, let's convert 38° 35' to decimal degrees:
38° 35' = 38 + (35/60) = 38.5833°
Now we can calculate angle A:
A = 180° - 38.5833° - 90°
A ≈ 51.4167°
So, angle A is approximately 51.42°.
In summary:
Side c ≈ 6.434 mi
Side b ≈ 5.066 mi
Angle A ≈ 51.42°