Questions: Understanding trigonometric ratios through similar right triangles The three right triangles below are similar. The acute angles angle A, angle D, and angle X are all approximately measured to be 41.1 degrees. The side lengths for each triangle are as follows. Note that the triangles are not drawn to scale. (a) For each triangle, find the ratio of the length of the side adjacent to 41.1 degrees to the length of the hypotenuse. Note that "side adjacent" cannot also refer to the hypotenuse. Round your answers to the nearest hundredth. AB / AC = DE / DF = XY / XZ = (b) Use the ALEKS Calculator to find sin 41.1 degrees, cos 41.1 degrees, and tan 41.1 degrees. Round your answers to the nearest hundredth. sin 41.1 degrees = cos 41.1 degrees = tan 41.1 degrees = (c) Which trigonometric function gives each ratio of sides in part (a)? sine cosine tangent none of these

Understanding trigonometric ratios through similar right triangles The three right triangles below are similar. The acute angles angle A, angle D, and angle X are all approximately measured to be 41.1 degrees. The side lengths for each triangle are as follows. Note that the triangles are not drawn to scale. (a) For each triangle, find the ratio of the length of the side adjacent to 41.1 degrees to the length of the hypotenuse. Note that "side adjacent" cannot also refer to the hypotenuse. Round your answers to the nearest hundredth. AB / AC = DE / DF = XY / XZ = (b) Use the ALEKS Calculator to find sin 41.1 degrees, cos 41.1 degrees, and tan 41.1 degrees. Round your answers to the nearest hundredth. sin 41.1 degrees = cos 41.1 degrees = tan 41.1 degrees = (c) Which trigonometric function gives each ratio of sides in part (a)? sine cosine tangent none of these
Transcript text: Understanding trigonometric ratios through similar right triangles The three right triangles below are similar. The acute angles $\angle A, \angle D$, and $\angle X$ are all approximately measured to be $41.1^{\circ}$. The side lengths for each triangle are as follows. Note that the triangles are not drawn to scale. (a) For each triangle, find the ratio of the length of the side adjacent to $41.1^{\circ}$ to the length of the hypotenuse. Note that "side adjacent" cannot also refer to the hypotenuse. Round your answers to the nearest hundredth. \[ \frac{A B}{A C}= \] $\square$ \[ \frac{D E}{D F}= \] $\square$ \[ \frac{X Y}{X Z}= \] $\square$ (b) Use the ALEKS Calculator to find $\sin 41.1^{\circ}, \cos 41.1^{\circ}$, and $\tan 41.1^{\circ}$. Round your answers to the nearest hundredth. \[ \sin 41.1^{\circ}= \] $\square$ \[ \cos 41.1^{\circ}= \] $\square$ \[ \tan 41.1^{\circ}= \] $\square$ (c) Which trigonometric function gives each ratio of sides in part (a)? sine cosine tangent none of these
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Solution

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Solution Steps

Step 1: Calculate the ratio of the side adjacent to 41.1° to the hypotenuse for each triangle
  • For triangle ABC: \[ \frac{AB}{AC} = \frac{5.5 \, \text{m}}{7.3 \, \text{m}} \approx 0.75 \]
  • For triangle DEF: \[ \frac{DE}{DF} = \frac{22 \, \text{m}}{29.2 \, \text{m}} \approx 0.75 \]
  • For triangle XYZ: \[ \frac{XY}{XZ} = \frac{33 \, \text{m}}{43.8 \, \text{m}} \approx 0.75 \]
Step 2: Use the calculator to find sin(41.1°), cos(41.1°), and tan(41.1°)
  • Using a calculator: \[ \sin(41.1°) \approx 0.66 \] \[ \cos(41.1°) \approx 0.75 \] \[ \tan(41.1°) \approx 0.87 \]
Step 3: Determine which trigonometric function gives each ratio of sides in part (a)
  • The ratio of the side adjacent to the hypotenuse is given by the cosine function.

Final Answer

  1. \[ \frac{AB}{AC} \approx 0.75, \frac{DE}{DF} \approx 0.75, \frac{XY}{XZ} \approx 0.75 \]
  2. \[ \sin(41.1°) \approx 0.66, \cos(41.1°) \approx 0.75, \tan(41.1°) \approx 0.87 \]
  3. Cosine
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