at this point, the moon is in the first quarter, lined up with earth, and the angle created from the earth…

at this point, the moon is in the first quarter, lined up with earth, and the angle created from the earth, to the moon, to the sun is a perpendicular angle. the distance at this time from the sun to the moon is 150 million km. the distance from the moon to the earth is about .384 million km. you need to find the distance from the earth to the sun.\n\nyou will also need to list the following information, for future reference:\n- all side lengths\n- all angle measures\n- all trig functions (sin, cos, tan, csc, sec, cot) for one of the following:\n - the angle from moon, earth, sun\n - the angle from earth, sun, moon

at this point, the moon is in the first quarter, lined up with earth, and the angle created from the earth, to the moon, to the sun is a perpendicular angle. the distance at this time from the sun to the moon is 150 million km. the distance from the moon to the earth is about .384 million km. you need to find the distance from the earth to the sun.\n\nyou will also need to list the following information, for future reference:\n- all side lengths\n- all angle measures\n- all trig functions (sin, cos, tan, csc, sec, cot) for one of the following:\n - the angle from moon, earth, sun\n - the angle from earth, sun, moon

Answer

Explanation:

Step1: Identify the right - triangle

The positions of the Earth, Moon, and Sun form a right - triangle. Let the distance from the Sun to the Moon be $a = 150$ million km, the distance from the Moon to the Earth be $b=0.384$ million km, and the distance from the Earth to the Sun be $c$. By the Pythagorean theorem, $c=\sqrt{a^{2}+b^{2}}$.

Step2: Calculate the distance from the Earth to the Sun

$c=\sqrt{(150)^{2}+(0.384)^{2}}=\sqrt{22500 + 0.147456}=\sqrt{22500.147456}\approx150$ million km.

Step3: List side lengths

The side lengths are: distance from Sun to Moon $a = 150$ million km, distance from Moon to Earth $b = 0.384$ million km, distance from Earth to Sun $c\approx150$ million km.

Step4: List angle measures

One angle is $90^{\circ}$ (the angle from the Earth, to the Moon, to the Sun). Let $\theta$ be the angle from the Moon, Earth, Sun. $\sin\theta=\frac{a}{c}\approx1$, $\theta\approx90^{\circ}$. The third angle (from Earth, Sun, Moon) is approximately $0^{\circ}$.

Step5: List trig - functions for the angle from Moon, Earth, Sun

Let $\theta$ be the angle from Moon, Earth, Sun. $\sin\theta=\frac{150}{150}=1$, $\cos\theta = 0$, $\tan\theta$ is undefined, $\csc\theta = 1$, $\sec\theta$ is undefined, $\cot\theta=0$.

Answer:

Distance from Earth to Sun: approximately 150 million km Side lengths: Sun - Moon: 150 million km, Moon - Earth: 0.384 million km, Earth - Sun: approximately 150 million km Angle measures: 90°, approximately 90°, approximately 0° Trig functions for angle from Moon, Earth, Sun: $\sin\theta = 1$, $\cos\theta = 0$, $\tan\theta$ is undefined, $\csc\theta = 1$, $\sec\theta$ is undefined, $\cot\theta = 0$