https://r2.erweima.ai/mathbot/image/1726536730473zisdrw3r.jpg

In the image you provided, there is a right triangle with the following information:

- Angle \( B = 25^\circ \)
- Side \( b = 10 \) (opposite angle \( B \))
- Side \( a \) (adjacent to angle \( B \)) is unknown
- Side \( c \) (the hypotenuse) is also unknown
- Angle \( \alpha \) (complementary to \( B \)) is unknown, but since it's a right triangle, \( \alpha = 90^\circ - 25^\circ \)

Let's calculate the missing sides and angle.

### 1. Calculate angle \( \alpha \):
Since the sum of the angles in a triangle is \( 180^\circ \) and one angle is \( 90^\circ \) (right angle), we can compute \( \alpha \):

\[
\alpha = 90^\circ - 25^\circ = 65^\circ
\]

### 2. Use trigonometric ratios to find side \( a \) (adjacent to \( B \)):
We can use the tangent function, which relates the opposite and adjacent sides:
\[
\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a}
\]
Rearranging the equation to solve for \( a \):
\[
a = \frac{b}{\tan(B)}
\]

### 3. Use the sine function to find side \( c \) (the hypotenuse):
\[
\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c}
\]
Rearranging the equation to solve for \( c \):
\[
c = \frac{b}{\sin(B)}
\]

Let me calculate these values for you.Here are the calculated values for the triangle:

1. \( \alpha = 65^\circ \)
2. \( a \approx 21.45 \) (adjacent side to angle \( B \))
3. \( c \approx 23.66 \) (the hypotenuse)

Would you like more details on these calculations, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How do we apply the sine rule in non-right triangles?
2. What other trigonometric functions can we use to find the sides of a triangle?
3. How can we calculate the area of a right triangle?
4. Can we use the Pythagorean theorem to verify the results here?
5. What are some real-life applications of trigonometry in measuring distances?

**Tip:** Always ensure your calculator is set to degrees (or radians, depending on the context) when using trigonometric functions.

Given a right triangle with angle B = 25 degrees and side b = 10, calculate the unknown sides and angles.

Learn how to solve a right triangle with a given angle B = 25° and side b = 10. The solution involves using trigonometric functions such as tangent and sine to find the unknown sides and angle.

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