Questions: Trigonometric Functions Understanding trigonometric ratios through similar right triangles The three right triangles below are similar. The acute angles angle L, angle R, and angle Z are all approximately measured to be 62.4 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 opposite 62.4 degrees to the length of the side adjacent to 62.4 degrees. Note that "side adjacent" cannot also refer to the hypotenuse. Round your answers to the nearest hundredth. JK/KL= square PQ/QR= square XY/YZ= square (b) Use the ALEKS Calculator to find sin 62.4 degrees, cos 62.4 degrees, and tan 62.4 degrees. Round your answers to the nearest hundredth. sin 62.4 degrees= square cos 62.4 degrees= square tan 62.4 degrees= square (c) Which trigonometric function gives each ratio of sides in part (a)? sine cosine tangent none of these

Trigonometric Functions Understanding trigonometric ratios through similar right triangles The three right triangles below are similar. The acute angles angle L, angle R, and angle Z are all approximately measured to be 62.4 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 opposite 62.4 degrees to the length of the side adjacent to 62.4 degrees. Note that "side adjacent" cannot also refer to the hypotenuse. Round your answers to the nearest hundredth. JK/KL= square PQ/QR= square XY/YZ= square (b) Use the ALEKS Calculator to find sin 62.4 degrees, cos 62.4 degrees, and tan 62.4 degrees. Round your answers to the nearest hundredth. sin 62.4 degrees= square cos 62.4 degrees= square tan 62.4 degrees= square (c) Which trigonometric function gives each ratio of sides in part (a)? sine cosine tangent none of these
Transcript text: Trigonometric Functions Understanding trigonometric ratios through similar right triangles The three right triangles below are similar. The acute angles $\angle L, \angle R$, and $\angle Z$ are all approximately measured to be $62.4^{\circ}$. 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 opposite $62.4^{\circ}$ to the length of the side adjacent to $62.4^{\circ}$. Note that "side adjacent" cannot also refer to the hypotenuse. Round your answers to the nearest hundredth. $\frac{J K}{K L}=$ $\square$ \[ \frac{P Q}{Q R}= \] $\square$ \[ \frac{X Y}{Y Z}= \] $\square$ (b) Use the ALEKS Calculator to find $\sin 62.4^{\circ}, \cos 62.4^{\circ}$, and $\tan 62.4^{\circ}$. Round your answers to the nearest hundredth. \[ \sin 62.4^{\circ}= \] $\square$ \[ \cos 62.4^{\circ}= \] $\square$ \[ \tan 62.4^{\circ}= \] $\square$ (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 opposite to the side adjacent for each triangle

For triangle \( \triangle JKL \):

  • Opposite side (JK) = 19.73 cm
  • Adjacent side (KL) = 9.15 cm \[ \frac{JK}{KL} = \frac{19.73}{9.15} \approx 2.16 \]

For triangle \( \triangle PQR \):

  • Opposite side (PQ) = 78.92 cm
  • Adjacent side (QR) = 36.6 cm \[ \frac{PQ}{QR} = \frac{78.92}{36.6} \approx 2.16 \]

For triangle \( \triangle XYZ \):

  • Opposite side (XY) = 118.38 cm
  • Adjacent side (YZ) = 54.9 cm \[ \frac{XY}{YZ} = \frac{118.38}{54.9} \approx 2.16 \]
Step 2: Use the calculator to find trigonometric values

Using a calculator: \[ \sin 62.4^\circ \approx 0.88 \] \[ \cos 62.4^\circ \approx 0.47 \] \[ \tan 62.4^\circ \approx 1.88 \]

Step 3: Identify the trigonometric function for the given ratios

The ratio of the side opposite to the side adjacent is the definition of the tangent function. \[ \tan 62.4^\circ = \frac{\text{opposite}}{\text{adjacent}} \]

Final Answer

  1. The ratios are:

    • \( \frac{JK}{KL} \approx 2.16 \)
    • \( \frac{PQ}{QR} \approx 2.16 \)
    • \( \frac{XY}{YZ} \approx 2.16 \)

  2. Trigonometric values:

    • \( \sin 62.4^\circ \approx 0.88 \)
    • \( \cos 62.4^\circ \approx 0.47 \)
    • \( \tan 62.4^\circ \approx 1.88 \)

  3. The trigonometric function for the given ratios is:

    • Tangent
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