Suppose Triangle ABC has right angle C. Find the measures of the other sides to the nearest whole number if...?
the measure of angle A is 32 degrees, and the measure of AB is 42.
For one side, I got 23, and for the other side, I got 35. Am I right?
correct
thanks
Check rounding--I got slightly different values that rounded to different numbers.
Given:
C = 90 degrees
A = 32 degrees
AB = 42 units
Therefore, B = 180 - 90- 32 = 58 degrees
Sine rule:
BC/sin(A) = AC/sin(B) = AB/sin(C)
BC/sin(32) = AC/sin(58) = 42/sin(90)
BC/(0.5299) = AC/(0.8480) = 42/(1.0)
BC = (0.5299)(42) = 22.26 round to 22
AC = (0.8480)(42) = 35.62 round to 36
I used an online graphing calculator..so maybe it's different like that?
To determine the measures of the other sides of a right triangle, we can use trigonometric ratios. Specifically, in a right triangle, we can use the sine, cosine, or tangent ratios.
In this case, we are given that angle A is 32 degrees, and side AB (opposite angle C) has a length of 42 units. To find the measures of the other sides, we need to determine whether we are looking for the lengths of the adjacent side (BC) or the opposite side (AC).
Let's assume that side BC is the adjacent side, and side AC is the opposite side. To find the length of side BC, we can use the cosine ratio:
cos(A) = BC / AB
cos(32°) = BC / 42
To find BC, we can rearrange the equation:
BC = cos(32°) × 42
Using a calculator or a trigonometric table, we can calculate the value of cos(32°) ≈ 0.8480. Therefore:
BC ≈ 0.8480 × 42 ≈ 35.616
So, if we round this to the nearest whole number, BC ≈ 36.
Now let's find the length of side AC using the sine ratio:
sin(A) = AC / AB
sin(32°) = AC / 42
AC = sin(32°) × 42
Using a calculator or a trigonometric table, we can calculate the value of sin(32°) ≈ 0.5299. Therefore:
AC ≈ 0.5299 × 42 ≈ 22.258
Rounding this to the nearest whole number, AC ≈ 22.
So, the measures of the other sides to the nearest whole number are BC ≈ 36 and AC ≈ 22.
Note: It's important to note that your initial calculations of 23 and 35 were close, but not exact, due to possible rounding errors.