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 asks us to find angle M in a right-angled triangle LMN, given that angle L is π/2 and the lengths of sides n and l are 3.1 and 8.4 respectively. We will use trigonometric relationships to calculate the angle and present the answer in radians, rounded to two decimal places.
Answer
Angle \( M \) is approximately \( 0.36 \) radians.
Answer for screen readers
The measure of angle ( M ) is approximately ( 0.36 ) radians.
Steps to Solve
- Identify Given Information
We have a right-angled triangle LMN where angle ( L = \frac{\pi}{2} ). The lengths of sides ( n ) and ( l ) are given as ( n = 3.1 ) and ( l = 8.4 ).
- Use Trigonometric Relationships
To find angle ( M ), we can use the sine, cosine, or tangent ratios. Here, we will use the tangent:
[ \tan(M) = \frac{\text{opposite}}{\text{adjacent}} = \frac{n}{l} = \frac{3.1}{8.4} ]
- Calculate the Tangent Ratio
Now, calculate the value of ( \tan(M) ):
[ \tan(M) = \frac{3.1}{8.4} \approx 0.3690 ]
- Find the Angle using Arctangent
To find angle ( M ), we take the arctangent:
[ M = \tan^{-1}(0.3690) ]
Using a calculator, we find:
[ M \approx 0.3575 \text{ radians} ]
- Round the Answer
Finally, we round our answer to two decimal places:
[ M \approx 0.36 \text{ radians} ]
The measure of angle ( M ) is approximately ( 0.36 ) radians.
More Information
Angle ( M ) in triangle LMN gives us a perspective on the relationship between the sides in a right triangle, illustrating how trigonometric functions can be used to determine angles based on side lengths.
Tips
- Miscalculating the Tangent Ratio: Ensure that you divide the lengths correctly.
- Incorrectly Using the Wrong Trigonometric Function: Make sure to correctly identify which sides correspond to "opposite" and "adjacent."
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