https://file.aiquickdraw.com/mathbot/file/1728907246840m8we4wuh.jpg

We are given a right triangle \( \triangle GEF \), where:

- \( \angle G \) is \( 40^\circ \),
- \( \angle F \) is \( 90^\circ \),
- \( GE = 9 \), and
- \( GF = x \), which we need to solve for.

Since we are dealing with a right triangle, we can use trigonometric ratios. Specifically, the sides involved are:

- The side opposite \( \angle G \) is \( GE = 9 \),
- The hypotenuse is \( GF = x \).

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

In this case:
\[
\sin(40^\circ) = \frac{9}{x}
\]

To solve for \( x \), rearrange the equation:
\[
x = \frac{9}{\sin(40^\circ)}
\]

Now, let's calculate:

\[
x = \frac{9}{\sin(40^\circ)} \approx \frac{9}{0.6428} \approx 14.0
\]

Thus, \( x \approx 14.0 \) when rounded to the nearest tenth.

Would you like further details on this? Here are some related questions:

1. How do you calculate the length of a side using cosine?
2. How would the solution change if we were solving for the adjacent side?
3. What is the definition of the tangent function in trigonometry?
4. How can we use the Pythagorean theorem to check the result?
5. How do you convert between degrees and radians?

**Tip:** Remember to always check if the triangle is a right triangle before using trigonometric ratios.

Solve for x. Round to the nearest tenth, if necessary. Given a right triangle GEF, where: - Angle G is 40°, - Angle F is 90°, - GE = 9, and - GF = x.

Learn how to solve for x in a right triangle where Angle G is 40°, Angle F is 90°, and GE = 9. Using trigonometric ratios, we solve for the hypotenuse GF. This problem covers key concepts like the sine function and is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation