For a circumpolar star, the complementary angle of the observer's latitude plus the complementary angle of the declination must be less than 90 degrees.
12.
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to " co-".
13.
The complementary angle of 70?( from the sun to the pole ) is added or subtracted from the observer's latitude to find the upper and lower culminations:
14.
Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless ? = 45?( a rectangular hyperbola ).
15.
The image shows complementary angles 27-32 and 63-58 respectively sharing the same values for XZ, Aside from the special case 45 degree angle which is its own complementary angle, other acute angles and their complementary angles share no value for the same function, keeping it simple here with first quadrant acute angles.
16.
The image shows complementary angles 27-32 and 63-58 respectively sharing the same values for XZ, Aside from the special case 45 degree angle which is its own complementary angle, other acute angles and their complementary angles share no value for the same function, keeping it simple here with first quadrant acute angles.
17.
The image shows complementary angles 27-32 and 63-58 respectively sharing the same values for XZ, Aside from the special case 45 degree angle which is its own complementary angle, other acute angles and their complementary angles share no value for the same function, keeping it simple here with first quadrant acute angles.
18.
Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e . g ., such that "'a "'?" "'b "'cos " ? "'} }, which can be determined by the pattern of the scalar product to be equal to:
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