Complete the table by evaluating appropriate ratios.
C
c
What can you say about the ratios looking at the table?
A
a
See solution.
B
b
See solution.
C
c
In any triangle, the ratio of the sine of an angle to the length of the leg opposite to this angle is constant.
Practice makes perfect
a
We are asked to draw three triangles, one acute, one obtuse, and one right. We will name these triangles ABC, MNP, and XYZ.
Next we will measure all side lengths using a ruler and find all the angles measures using a protractor.
b
In this part we are asked to complete the given table using the measures we found. Let's substitute appropriate side lengths and evaluate appropriate trigonometric ratios using a calculator.
Triangle
Ratios
ABC
sin A/BC=sin 46^(∘)/2.2≈ 0.3
sin B/AC=sin 69^(∘)/3≈ 0.3
sin C/AB=sin 65^(∘)/2.8≈ 0.3
MNP
sin M/NP=sin 103^(∘)/5.3≈ 0.2
sin N/PM=sin 28^(∘)/2.5≈ 0.2
sin P/MN=sin 49^(∘)/4.1≈ 0.2
XYZ
sin X/YZ=sin 90^(∘)/3.6≈ 0.28
sin Y/ZX=sin 56^(∘)/3≈ 0.28
sin Z/XY=sin 34^(∘)/2≈ 0.28
c
Looking at the table we can see that approximate ratios are constant for each of the triangles. Therefore, we can assume that in any triangle the ratio of the sine of an angle to the length of the leg opposite to this angle is constant.
