Find the lengths of the unknown sides in the right triangle shown to the right.
35 degrees
122
b
B
c
A
C
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Part 1
bequals
  
enter your response here ​(Simplify your answer. Type an integer or decimal rounded to the nearest tenth as​ needed.)
cequals
  
enter your response here ​(Simplify your answer. Type an integer or decimal rounded to the nearest tenth as​ needed.)

To find the lengths of the unknown sides of a right triangle, we can use trigonometric ratios (sine, cosine, and tangent) based on the given angle and the known side. In this case, we are given an angle of \( 35^\circ \) and the length of one side, \( 122 \), which we assume to be the adjacent side.

### Step-by-Step Process

Let’s label the triangle:

- \( A \): Angle of \( 35^\circ \)
- \( C \): Right angle (90 degrees)
- \( a = 122 \): Adjacent side (side next to \( 35^\circ \))
- \( b \): Opposite side (side opposite \( 35^\circ \))
- \( c \): Hypotenuse (longest side)

### 1. Finding \( b \) (Opposite side)
We can use the tangent function, as \( \tan(\theta) \) is the ratio of the opposite side to the adjacent side:
\[
\tan(35^\circ) = \frac{b}{122}
\]
Solving for \( b \):
\[
b = 122 \times \tan(35^\circ)
\]
Using a calculator:
\[
b \approx 122 \times 0.7002 = 85.4
\]

### 2. Finding \( c \) (Hypotenuse)
We can use the cosine function, as \( \cos(\theta) \) is the ratio of the adjacent side to the hypotenuse:
\[
\cos(35^\circ) = \frac{122}{c}
\]
Solving for \( c \):
\[
c = \frac{122}{\cos(35^\circ)}
\]
Using a calculator:
\[
c \approx \frac{122}{0.8192} = 148.9
\]

### Final Answer:
- \( b \approx 85.4 \)
- \( c \approx 148.9 \)

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions:
1. How do you determine which trigonometric function to use for each side?
2. What is the sine function, and how could we use it to find \( b \)?
3. How would the calculation change if the given angle were \( 55^\circ \) instead of \( 35^\circ \)?
4. Can you solve for the missing sides if both angles other than the right angle are unknown?
5. How would you calculate the area of this triangle?

**Tip**: Always double-check the mode of your calculator (degrees or radians) when dealing with trigonometric functions to ensure accurate results.

Find the lengths of the unknown sides in the right triangle shown to the right. Given an angle of 35 degrees and the length of one side (122), find the lengths of sides b (opposite) and c (hypotenuse).

Learn how to find the lengths of the unknown sides of a right triangle using trigonometric ratios, given a 35-degree angle and one side length (122). This solution covers key trigonometry concepts such as the tangent and cosine functions. Suitable for students in Grades 9-12.

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