Trigonometry Fundamentals

Trigonometry

• sin A cos B – cos A sin B is equivalent to sin (A – B).
• The angular distance of a point on the terrestrial sphere from the north pole is called coaltitude.
• Csc 520° is equal to Csc 20°.
• 0.984 is the sine of 820°.
• The logarithm of a negative number is imaginary.
• The sum of the squares of the sine and cosine of an angle is 1.
• The logarithm of a number to the base (2.7182....) is called a Napierian logarithm.
• When a number is written in logarithmic form, the characteristic is equal to the exponent of 10.
• Napierian logarithms have a base closest to 2.72.
• A logarithm of 1 to any base is zero.
• Sin (270 + β ) is equal to - Cos β.
• The sum of the angles in an octant spheric triangle is 270°.
• The median of a triangle connects the vertex and the midpoint of the opposite side - the intersection of these medians is called the centroid.
• The altitudes of the sides of a triangle intersects at the point known as the orthocenter.
• The angle of elevation is the angle which the line of sight to the object makes with the horizontal, above the eye of the observer.
• Log M – Log N is equal to Log M/N.
• The other form of logₐ N = b is N = a^b.
• The point of concurrency of the altitude of the triangle is the orthocenter.
• The point of concurrency of the perpendicular bisector of the sides of the triangle is the circumcenter.
• The point of concurrency of the angle bisector of the triangle is called the incenter.
• The logarithm of the reciprocal of N is called the cologarithm of N.
• The inverse function of a logarithm is the antilogarithm.
• The cologarithm of a number is the negative of the logarithm of a number.
• The first table logarithm with 10 as base was developed in 1615 by Henry Briggs.
• John Napier invented logarithms in 1614.
• The number logₐ b is called the modulus of the system of a base a with respect to the system of base b.
• The base of a Napierian logarithm is e.
• Log x = 0.434 ln x.
• ln x = 2.303 log x.
• 1 cannot be a base of a logarithm.
• The integral part of a common logarithm is the characteristic.
• The mantissa of a logarithm is a positive value or zero.
• For 0 < x < 1, ln x is negative.
• If 1 < N < 10, then 1 < log N < 2.
• To change logₐ to logb N, multiply logₐ N by logb a.
• The numbers logₐ b and log♭ a are reciprocals of each other.
• A logarithm using 10 as a base is a common logarithm.
• The logarithm of a product is the sum of the logarithms, while the logarithm of a quotient is the difference of the logarithms.
• When a logarithm is expressed as an integer plus a decimal (between 0 and 1), the integer is called the characteristics.
• If the characteristics of a logarithm is 3, the number is between 1000 and 10000.
• The characteristics of the common logarithm of a number greater than 1is zero or positive.
• The characteristic equals the exponent of 10 when the number is written in scientific notation.
• The logarithm of base 2 (denotes as lb) is called a binary logarithm.
• If the unknown occurs as an exponent in a conditional equation, the best way to solve it is by taking the logarithm of both sides.
• Angles of rotation with the same initial side and terminal side are co-terminal angles.
• An angle equal to one revolution of 360° is a perigon.
• The angle which the line of sight to the object makes with the horizontal above the eye of an observer is the angle of elevation.
• A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides of the same point inside the given triangle is a Pedal triangle.
• A pedal triangle is the triangle with minimum perimeter but maximum area inscribed in another triangle.
• A right triangle whose length of sides could be expressed as the ratio of integral units is a primitive triangle.
• A triangle with no side equal is known as a Scalene triangle.
• If two triangles have congruent bases, then the ratio of their areas equals the ratio of the lengths of their altitudes.
• In an isosceles right triangle, the hypotenuse is √2 times as long as each of the legs.
• Sides are not a secondary part of a triangle.
• The sum of two sides is less than the third side is not a property of a triangle.
• Given the sides of a triangle as 3m and 5m, then the 3rd side is from 3m to 7m.
• A straight line from the vertex of a triangle to the midpoint of the opposite side is known as the median.
• Three or more lines which have one point in common are said to be coplanar: This is a false statement.
• The case of the solution of the triangle in the plane where the given data leads to two solutions is the ambiguous case.
• The most proved theorem in mathematics is the Pythagorean theorem.
• The least proved theorem in mathematics is Fermat’s Last Theorem.
• Equations used for checking the solution to a plane triangle using law of sine's are Mollweide’s equations.
• Napier’s rule states that the sine of any middle part is equal to the product of the cosine of the opposite parts.
• Napier’s rule states that the sine of any middle part is equal to the product of the tangent of the adjacent parts.
• Ten formulas may be derived by using Napier’s rules.
• The sum of all interior angles in a spherical triangle is always greater than 180° but less than 540°.
• The maximum value for longitude is 180°.
• The maximum value for the latitude is 90°.
• If R is the radius of a sphere and E is a spherical excess (in radians), then the area of a spherical triangle is R²E.
• One minute of the great circle arc on the surface of the earth is equivalent to 1 nautical mile.
• A spherical triangle with all angles equal to a right triangle is called a birectangular spherical triangle.
• A spherical triangle with at least one side as a quarter of a great circle is a trirectangular spherical triangle.
• One of the two great circles intersecting at right angles at the piles and dividing equinoctial points and ecliptic into 4 parts is called colure.
• The radius of the earth used in spherical trigonometry is 3959 statute miles.
• The difference between a nautical mile and a statute mile is 800 feet.
• Manila has a longitude of 121°05'E; the time difference between Manila and Greenwich, England is 8 hours 4 minutes.
• The earth is divided into 24 time zones.
• In a spherical triangle, two angles (or sides) are on the same species if they are both between 0° and 90° or both between 90° and 180°.
• A hemisphere has 360° spherical degrees.
• The sum of all interior angles of a spherical triangle is 360° it is false about spherical trigonometry.
• The sum of the sides of a spherical triangle is always less than 360°.
• The sum of any two angles of a spherical triangle is less than 180° + the third angle
• Latitude refers to the angular distance from the equator measured along a meridian.
• Longitude refers to the angle at either pole between the meridian passing through a point and some fixed meridian, known as the prime meridian.
• Meridian refers to half of a great circle terminated by the north pole and south pole.
• When the hypotenuse of a right spherical triangle is less than 90°, then the two legs are on the same quadrant.
• When the hypotenuse of a right spherical triangle is greater than 90°, one leg is on the first quadrant and the other is on the second quadrant.
• Zenith is the point where a ray from the center of the Earth through an observer's position intersects the celestial sphere.
• Nadir is the point that is diametrically opposite the zenith
• The great circles through the north and south celestial poles are called Hour circles and Celestial meridians.
• An oblique equilateral parallelogram is a rhombus.
• Mil is a unit of angle and length.