Given a right angled triangle LMN with angle L = π/2 and sides n = 3.1 and l = 8.4, find angle M. Give your answer in radians to 2 decimal places.
Understand the Problem
The question is asking us to find the measure of angle M in a right-angled triangle where one of the angles (L) is given as π/2. We know the lengths of two sides (n and l) of the triangle, and we will use trigonometric relationships to calculate angle M.
Answer
$$ M = \arcsin\left(\frac{n}{l}\right) $$
Answer for screen readers
The measure of angle $M$ is given by the equation:
$$ M = \arcsin\left(\frac{n}{l}\right) $$
Steps to Solve
- Identify the triangle's angles and sides
In a right-angled triangle, one angle is always $90^\circ$ or $\frac{\pi}{2}$ radians. Given that angle $L = \frac{\pi}{2}$, the other two angles must add up to $\frac{\pi}{2}$ radians, meaning $M + N = \frac{\pi}{2}$.
- Use the sine function to find angle M
If we know the lengths of the two sides, we can use the sine function. Assuming side $n$ is the opposite side to angle $M$, and side $l$ is the hypotenuse, we write:
$$ \sin(M) = \frac{n}{l} $$
- Solve for angle M
To find angle $M$, we will take the inverse sine (arcsin) of both sides:
$$ M = \arcsin\left(\frac{n}{l}\right) $$
This gives us the angle $M$ in radians.
The measure of angle $M$ is given by the equation:
$$ M = \arcsin\left(\frac{n}{l}\right) $$
More Information
Angle $M$ represents one of the two acute angles in the right triangle. The sine function relates the opposite side to the hypotenuse, which is fundamental in trigonometry. This relationship allows us to calculate angles based on side lengths.
Tips
- Forgetting that the triangle must adhere to the properties of a right-angled triangle, particularly the relationship between the angles.
- Confusing opposite and adjacent sides when using trigonometric functions.
- Not ensuring that the hypotenuse is indeed the longest side.
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