Right △EFG has its right angle at F, EG=8, and FG=6.
What is the value of the trigonometric ratio of an angle of the triangle? Match the trigonometric ratio with its value.
sec G=
cot G=
sin E=
Choices: 3/4, 4/3, √7/3, √7/8, 3√7/7, 4√7/7
To find the value of the trigonometric ratios, let's begin by labeling the sides of the triangle.
In triangle EFG:
- EG = 8 (the side opposite angle E)
- FG = 6 (the side adjacent to angle E)
- The right angle is at F
Now, let's determine the missing side values using the Pythagorean theorem.
Since the right angle is at F, we can find the length of the third side (EF) using the Pythagorean theorem:
EF^2 = EG^2 + FG^2
EF^2 = 8^2 + 6^2
EF^2 = 64 + 36
EF^2 = 100
EF = √100
EF = 10
Now, let's calculate the trigonometric ratios:
1. sec G: secant is the reciprocal of cosine.
To find sec G, we need to use the cosine ratio.
cos G = adjacent/hypotenuse = FG/EF = 6/10 = 3/5
sec G = 1/cos G = 1/(3/5) = 5/3
Therefore, sec G = 5/3.
2. cot G: cotangent is the reciprocal of tangent.
To find cot G, we need to use the tangent ratio.
tan G = opposite/adjacent = EG/FG = 8/6 = 4/3
cot G = 1/tan G = 1/(4/3) = 3/4
Therefore, cot G = 3/4.
3. sin E: sine is the ratio between the opposite side and the hypotenuse.
sin E = opposite/hypotenuse = EG/EF = 8/10 = 4/5
Therefore, sin E = 4/5.
Matching the trigonometric ratios with their values:
sec G = 5/3
cot G = 3/4
sin E = 4/5
Therefore, the choices for the trigonometric ratios and their values are:
sec G = 5/3 (option: 4/3)
cot G = 3/4 (option: 3/4)
sin E = 4/5 (option: √7/8)
from G, adjacent = GF, hypotenuse = EG
from E, opposite = GF, hypotenuse = EG
now just look up the definitions of the trig functions