Questions & Answers
Browse all 20 questions from the
Celestial Trigonometry and Pacific Navigation study set below.
Each question shows the correct answer — select a study format above to practice interactively.
1
In spherical trigonometry, what is the law used by Polynesian wayfinders to calculate the great-circle distance between two Pacific islands using celestial coordinates?
-
A
Law of Cosines for sides
-
B
Law of Sines
-
C
Pythagorean Theorem
-
D
Law of Tangents
2
The declination of the star Sirius, used by Pacific navigators for latitude determination, is approximately -16.7 degrees. What is its polar distance in a trigonometric coordinate system?
-
A
106.7 degrees
-
B
73.3 degrees
-
C
16.7 degrees
-
D
90 degrees
3
When calculating the zenith distance of a star at an island in the South Pacific, what trigonometric function relates the altitude of the star to the observer's latitude?
-
A
Sine
-
B
Cosine
-
C
Tangent
-
D
Secant
4
The IAU defined the astronomical unit (AU) based on the parallax of the Sun. What trigonometric value is used to calculate the distance to a planet given its parallax angle?
-
A
sin(p)
-
B
tan(p)
-
C
cot(p)
-
D
csc(p)
5
To determine the true azimuth of a celestial body, Pacific navigators utilized the spherical triangle formed by the pole, the zenith, and the star. Which law resolves this triangle?
-
A
Napier's Rules
-
B
Haversine Formula
-
C
Brahmagupta's Formula
-
D
Heron's Formula
6
The Earth's orbital eccentricity (e = 0.0167) affects the calculation of the Sun's position. What trigonometric identity is used in the equation of center to relate the true anomaly to the mean anomaly?
-
A
Taylor series expansion
-
B
Double angle formula
-
C
Product-to-sum identity
-
D
Inverse tangent
7
At the Mauna Kea Observatory, astronomers measure the angular diameter of distant stars. If the angle is theta, what trigonometric relationship calculates the physical diameter D at distance L?
-
A
D = L sin(theta/2)
-
B
D = 2L tan(theta/2)
-
C
D = L cos(theta)
-
D
D = L sec(theta)
8
In the context of the Pacific's night sky, the Southern Cross (Crux) is used for latitude. If the stars Gacrux and Acrux define a line, what trigonometric function describes the variation of their separation due to atmospheric refraction?
-
A
Tangent of the altitude
-
B
Sine of the azimuth
-
C
Cosine of the zenith
-
D
Secant of the refraction angle
9
What is the value of the parallax constant used in astronomical trigonometric distance measurements, defined as the angle subtended by one AU at a distance of one parsec?
-
A
1 arcsecond
-
B
1 radian
-
C
1 degree
-
D
1 arcminute
10
In planetary motion, Kepler's equation M = E - e sin E requires the conversion of what trigonometric parameter?
-
A
Eccentric anomaly
-
B
True anomaly
-
C
Mean anomaly
-
D
Phase angle
11
The 'star compass' used by traditional Pacific voyagers divides the horizon into segments. If there are 32 primary points, what is the trigonometric increment in degrees?
-
A
11.25
-
B
10.5
-
C
12.5
-
D
9.0
12
When observing a lunar eclipse from a Pacific island, the shadow of the Earth is used to calculate the Moon's distance. Which trigonometric ratio relates the Earth's radius to the shadow cone angle?
-
A
Sine
-
B
Cosine
-
C
Tangent
-
D
Cotangent
13
The angular separation between two planets viewed from Fiji is calculated using the spherical law of cosines. Given sides a and b and included angle C, what is the formula for side c?
-
A
cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)
-
B
cos(c) = sin(a)sin(b) + cos(a)cos(b)cos(C)
-
C
c = a + b - 2ab cos(C)
-
D
sin(c) = sin(a)cos(b) + cos(a)sin(b)
14
What trigonometric term describes the difference between the geocentric and heliocentric positions of a planet as observed from the Earth?
-
A
Parallactic angle
-
B
Phase angle
-
C
Elongation
-
D
Inclination
15
In spherical geometry, the sum of the angles of a triangle on a sphere is always greater than 180 degrees. This excess relates to the area of the triangle via which trigonometric function?
-
A
Sine
-
B
Cosine
-
C
Tangent
-
D
Secant
16
To calculate the solar altitude for a given time and date at a Pacific island, astronomers use the formula sin(h) = sin(phi)sin(delta) + cos(phi)cos(delta)cos(H). What does H represent?
-
A
Hour angle
-
B
Declination
-
C
Latitude
-
D
Zenith
17
The maximum elongation of Venus is the point where the Earth-Venus-Sun angle is 90 degrees. Using trigonometry, which side of the triangle is the sine of the elongation?
-
A
Earth-Sun distance
-
B
Earth-Venus distance
-
C
Venus-Sun distance
-
D
Venus radius
18
The obliquity of the ecliptic, approximately 23.4 degrees, requires what trigonometric adjustment when calculating the seasonal solar declination?
-
A
Sine conversion
-
B
Cosine integration
-
C
Tangent projection
-
D
Secant offset
19
For a stellar object passing through the observer's meridian at a Pacific island, what is the trigonometric relationship between declination (delta), latitude (phi), and zenith distance (z)?
-
A
z = phi - delta
-
B
z = phi + delta
-
C
z = delta - phi
-
D
z = 90 - (phi + delta)
20
When calculating the aberration of light, the shift in the apparent position of a star is a trigonometric function of the ratio between Earth's orbital velocity and what?
-
A
Speed of light
-
B
Escape velocity
-
C
Escape velocity
-
D
Angular momentum
