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6:56 124 Assignment Overview 3 DeltaMath Score: 0/4 Penalty: 1 off Identifying Trig Ratios (No Diagram) Question Video Examples In \( \triangle \mathrm{LMN} \), the measure of \( \angle \mathrm{N}=90^{\circ}, \mathrm{ML}=97 \), \( \mathrm{LN}=72 \), and \( \mathrm{NM}=65 \). What is the value of the tangent of \( \angle \mathrm{M} \) to the nearest hundredth? Answer Attempt 2 out of 2 \( \square \) Submit Answer Still Stuck? Copyright ©2025 DeltaMath.com All Rights Reserved. Terms \& Policies deltamath.com

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Feb 05,2025

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The tangent of \( \angle M \) is approximately 1.11.

Solución

To find the value of the tangent of \( \angle M \) in \( \triangle \mathrm{LMN} \), follow these steps: 1. **Identify the Sides:** - \( \angle \mathrm{N} \) is the right angle (\(90^\circ\)), so side \( \mathrm{ML} \) (97 units) is the **hypotenuse**. - The other two sides are: - \( \mathrm{LN} = 72 \) units - \( \mathrm{NM} = 65 \) units 2. **Determine the Relationship for \( \tan(\angle \mathrm{M}) \):** - **Opposite side to \( \angle \mathrm{M} \):** \( \mathrm{LN} = 72 \) units - **Adjacent side to \( \angle \mathrm{M} \):** \( \mathrm{NM} = 65 \) units 3. **Calculate the Tangent:** \[ \tan(\angle \mathrm{M}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{72}{65} \] 4. **Compute the Value:** \[ \tan(\angle \mathrm{M}) = \frac{72}{65} \approx 1.1077 \] Rounded to the nearest hundredth: \[ \tan(\angle \mathrm{M}) \approx 1.11 \] **Answer:** \(1.11\)

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The Deep Dive

In a right triangle, the tangent of an angle is calculated as the ratio of the length of the opposite side to the length of the adjacent side. In triangle LMN, to find \( \tan(\angle M) \), you would consider side LN as the opposite side (72) and side NM as the adjacent side (65). Thus, \( \tan(\angle M) = \frac{LN}{NM} = \frac{72}{65} \). Now, converting the fraction to a decimal gives approximately 1.1077. Rounding it to the nearest hundredth, you’ll get \( \tan(\angle M) \approx 1.11 \). So, don’t forget to always keep your triangles and their sides straight—it's essential for accurate calculations in trigonometry!

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