Question
Solve the right triangle ABC for all missing parts. Express angles in decimal degrees.
(Round to the nearest hundredth as needed.)
(Round to the nearest hundredth as needed.)
Ask by Bush Haynes. in the United States
Feb 02,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Angle
Solution
To solve the right triangle ABC, we have the following known values:
- Angle
- Side
(the hypotenuse)
We need to find:
- Angle
- Side
(opposite angle ) - Side
(adjacent to angle )
Step 1: Convert Angle A to Decimal Degrees
First, we convert angle
Step 2: Find Angle B
Since the sum of angles in a triangle is
Now, let’s calculate
Step 3: Calculate Angle B
Now, we will calculate angle
Step 4: Calculate Side a and Side b
Using the sine and cosine functions, we can find sides
Now, let’s calculate
I’ll perform these calculations now.
Calculate the value by following steps:
Calculate the value by following steps:
- step0: Calculate:
- step1: Subtract the numbers:
Calculate or simplify the expression.
Calculate the value by following steps: - step0: Calculate:
- step1: Simplify:
Calculate or simplify the expression.
Calculate the value by following steps: - step0: Calculate:
- step1: Simplify:
Step 2: Calculate Angle B
From our calculation:
Step 3: Calculate Side a and Side b
Now, we have:
- Side
(opposite angle ):
- Side
(adjacent to angle ):
Summary of Results
- Angle
- Side
- Side
Thus, the missing parts of triangle ABC are:
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Explain 
Simplify this solution Beyond the Answer
To find angle B, we can use the fact that the sum of angles in a triangle is 180 degrees.
First, convert angle A into decimal degrees:
Since we need to find angle B:
We can first apply the Law of Sines to find side a (opposite angle A):
Using this, we can calculate C:
To find the accurate angle B:
- Find c’s corresponding angle C using the Law of Sines.
- Substitute the computed angle A and side c into the equation.
Let’s start with C:
Rearranging for B involves numerical methods or sine values. However, without further calculations, we can derive angle B using numerical solvers or tools, approximating as follows:
Plugging values will yield approximately:
Considering all calculations:
- The remaining angle C can be derived similarly as noted above.
This procedure entails systematically embracing the relationships in triangles—geometry at its finest!
B is approximately 29.13° (rounded to nearest hundredths).
C can be computed as:
Now, we have all the angles:
- A ≈ 20.87°
- B ≈ 29.13°
- C ≈ 130.00°
Next, applying these values in context will significantly enhance your structural design’s accuracy! Keep in mind the importance of these relations, especially for engineers and architects—they rely heavily on precise angle measures to ensure structural integrity.
