Final answer:
The trigonometric ratios for angles E and G in a right triangle with sides 4, 5, and hypotenuse 13 can be determined using definitions of sine, cosine, and tangent, resulting in sin E = 5/13, tan E = 5/4, cos G = 4/13, cos E = 4/13, sin G = 4/13, and tan G = 13/5.
Step-by-step explanation:
Based on the provided information, we can use trigonometric identities and the Pythagorean theorem to find the unknown trigonometric ratios. From the given triangle with sides 4, 5, and hypotenuse 13, we can use the definitions of sine, cosine, and tangent:
- The sine of an angle (sin E) is the length of the opposite side divided by the hypotenuse.
- The cosine of an angle (cos G) is the length of the adjacent side divided by the hypotenuse.
- The tangent of an angle (tan E) is the length of the opposite side divided by the adjacent side.
In this right triangle, angle E's opposite side is 5, its adjacent side is 4, and the hypotenuse is 13. So we calculate:
- sin E = 5/13,
- tan E = 5/4,
- Since angle G would be complementary to angle E in our right triangle, we use the Pythagorean identity to find cos G, which is 4/13 because cos G = cos (90° - E) = sin E.
The ratios for the complementary angle G would be the reciprocals of the ones for E:
- cos E = 4/13,
- sin G = sin (90° - E) = cos E = 4/13,
- tan G = 13/5, which is the reciprocal of tan E.