Estimate the values to complete the table. Round to the nearest hundredth if necessary.

Question image

Understand the Problem

The question is asking to estimate values related to the angles and sides of a right triangle, represented by points P, Q, and R. Specifically, it involves calculating trigonometric ratios and rounding the results to the nearest hundredth. The triangle's measurements will be completed in a table format.

Answer

Angle R's values are: - Adjacent leg $\div$ Hypotenuse = 0.95 - Opposite leg $\div$ Hypotenuse = 0.31 - Opposite leg $\div$ Adjacent leg $\approx 3.06$
Answer for screen readers

The completed values for the table are:

  • For angle P:

    • Adjacent leg $\div$ Hypotenuse = 0.31
    • Opposite leg $\div$ Hypotenuse = 0.95
  • For angle R:

    • Adjacent leg $\div$ Hypotenuse = 0.95
    • Opposite leg $\div$ Hypotenuse = 0.31
    • Opposite leg $\div$ Adjacent leg $\approx 3.06$

Steps to Solve

  1. Calculate the trigonometric ratios for angle P

We have the adjacent leg divided by the hypotenuse (0.31) and the opposite leg divided by the hypotenuse (0.95).

For angle P, we can use the cosine and sine definitions:

  • $\cos(P) = \frac{\text{adjacent leg}}{\text{hypotenuse}}$
  • $\sin(P) = \frac{\text{opposite leg}}{\text{hypotenuse}}$

Using these ratios:

  • Hypotenuse = 1 (assuming it as 1 unit for simplicity)

From the given values:

  • The adjacent leg can be calculated as $0.31 \cdot \text{Hypotenuse} = 0.31 \cdot 1 = 0.31$
  • The opposite leg can be calculated as $0.95 \cdot \text{Hypotenuse} = 0.95 \cdot 1 = 0.95$

Next, we must determine angle R.

  1. Determine the trigonometric ratios for angle R

The triangle’s angle P and angle R are complementary, since the sum of angles in a right triangle is (90^\circ). Therefore:

$$ R = 90^\circ - P $$

Now we can write the trigonometric ratios for angle R:

  • For adjacent leg divided by hypotenuse: $\cos(R) = \frac{\text{opposite leg}}{\text{hypotenuse}} = \frac{0.95}{1} = 0.95$
  • For opposite leg divided by hypotenuse: $\sin(R) = \frac{\text{adjacent leg}}{\text{hypotenuse}} = \frac{0.31}{1} = 0.31$
  1. Calculate opposite leg divided by adjacent leg

To complete the opposite leg divided by adjacent leg ratio for angle R:

$$ \frac{\text{opposite leg}}{\text{adjacent leg}} = \frac{0.95}{0.31} $$

Calculating this gives approximately:

$$ \frac{0.95}{0.31} \approx 3.06 $$

  1. Final values for the table

Now we can summarize the values:

  • For angle P:

    • Adjacent leg $\div$ Hypotenuse = 0.31
    • Opposite leg $\div$ Hypotenuse = 0.95
  • For angle R:

    • Adjacent leg $\div$ Hypotenuse = 0.95
    • Opposite leg $\div$ Hypotenuse = 0.31
    • Opposite leg $\div$ Adjacent leg $\approx 3.06$

The completed values for the table are:

  • For angle P:

    • Adjacent leg $\div$ Hypotenuse = 0.31
    • Opposite leg $\div$ Hypotenuse = 0.95
  • For angle R:

    • Adjacent leg $\div$ Hypotenuse = 0.95
    • Opposite leg $\div$ Hypotenuse = 0.31
    • Opposite leg $\div$ Adjacent leg $\approx 3.06$

More Information

We found the necessary trigonometric ratios based on the definitions of sine and cosine, as well as understanding that the angles in a right triangle sum to (90^\circ). This problem requires knowledge of basic trigonometry to find relationships between the angles and sides.

Tips

  • Confusing the definitions of sine and cosine when calculating the ratios.
  • Not realizing that angles in a right triangle are complementary, which affects calculations for angle R.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser