TRIGONOMETRY

sin A cos B - cos A sin B is equivalent to

A. cos(A - B)

B. sin(A - B)

C. tan(A - B)

D. cos2(A - B)

ANSWER: B. sin(A - B)

The angular distance of a point on the terrestrial sphere from the north pole is called

A. coaltitude

B. latitude

C. altitude

D. codeclination

ANSWER: A. coaltitude

csc 520° is equal to

A. cos 20°

B. csc 20°

C. tan 45°

D. sin 20°

ANSWER: B. csc 20°

What is the sine of 820°?

A. 0.984

B. 0.866

C. -0.866

D. -0.5

ANSWER: A. 0.984

The logarithm of the negative number is

A. imaginary

B. irrational

C. real

D. rational

ANSWER: A. imaginary

The sum of the squares of the sine and cosine of an angle.

A. 0

B. 1

C. 2

D. 3

ANSWER: B. 1

The logarithm of a number to the base e (2.7182...) is called

A. Naperian logarithm

B. characteristic

C. mantissa

D. Briggsian logarithm

ANSWER: A. Naperian logarithm

The characteristic is equal to the exponent of 10, when the number is written in

A. exponential form

B. scientific notation

C. logarithmic form

D. irrational number

ANSWER: C. logarithmic form

Naperian logarithms have a base closest to which number?

A. 2.72

B. 2.82

C. 2.92

D. 10

ANSWER: A. 2.72

The logarithm of 1 to any base is

A. indeterminate

B. zero

C. infinity

D. one

ANSWER: B. zero

sin (270° + β) is equal to

A. -cos β

B. sin β

C. -sin β

D. cos β

ANSWER: A. -cos β

The sum of the angles in an octant spheric triangle is

A. 180°

B. 270°

C. 360°

D. 540°

ANSWER: B. 270°

The median of a triangle is the line connecting the vertex and the midpoint of the opposite side. For a given triangle, these medians intersects at a point which is called the

A. orthocenter

B. circumcenter

C. centroid

D. incenter

ANSWER: C. centroid

The altitudes of the sides of the triangle intersects at the point known as

A. orthocenter

B. circumcenter

C. incenter

D. centroid

ANSWER: A. orthocenter

The angle which the line of sight to the object makes with the horizontal is above the eye of an observer.

A. angle of depression

B. angle of elevation

C. acute angle

D. bearing

ANSWER: B. angle of elevation

log M - log N is equal to

A. log MN

B. log (M - N)

C. log M/N

D. log (N - M)

ANSWER: C. log M/N

The other form of logₐN = b is

A. N = bᵃ

B. N = aᵇ

C. N = ab

D. N = a/b

ANSWER: B. N = aᵇ

The point of concurrency of the altitude of the triangle.

A. orthocenter

B. centroid

C. metacenter

D. incenter

ANSWER: A. orthocenter

The point of concurrency of the perpendicular bisector of the sides of the triangle.

A. orthocenter

B. circumcenter

C. centroid

D. incenter

ANSWER: B. circumcenter

The point of concurrency of the angle bisector of the triangle is called

A. orthocenter

B. circumcenter

C. centroid

D. incenter

ANSWER: D. incenter

The logarithm of the reciprocal of N is called the _____ of N.

A. antilogarithm

B. cologarithm

C. natural logarithm

D. Briggsian logarithm

ANSWER: B. cologarithm

The inverse function of a logarithm is known as

A. antilogarithm

B. cologarithm

C. antiderivative

D. antecedent

ANSWER: A. antilogarithm

The cologarithm of a number is the _____ of the logarithm of a number.

A. positive

B. absolute value

C. negative

D. reciprocal

ANSWER: C. negative

The first table logarithms with 10 as base was developed in 1615 by

A. James Naismith

B. Henry Briggs

C. John Napier

D. John Wallis

ANSWER: B. Henry Briggs

Who invented logarithms in 1614?

A. John Wallis

B. Henry Briggs

C. John Napier

D. L'Hospital

ANSWER: C. John Napier

The number logab is called the _____ of the system of base a with respect to the system of base b.

A. coefficient

B. logarithm

C. modulus

D. exponent

ANSWER: C. modulus

Naperian logarithm has a base of

A. π

B. 10

C. 1

D. e

ANSWER: D. e

log x = _____ ln x.

A. 0.434

B. 10

C. 2.303

D. e

ANSWER: A. 0.434

ln x = _____ log x.

A. 0.434

B. 10

C. 2.303

D. e

ANSWER: C. 2.303

Which of the following cannot be a base for a logarithm?

A. 10

B. π

C. 1

D. e

ANSWER: C. 1

The integral part of a common logarithm is

A. 10

B. e

C. mantissa

D. characteristic

ANSWER: D. characteristic

The mantissa of a logarithm is a

A. positive value only

B. negative value only

C. positive value, negative value or zero

D. positive value or zero

ANSWER: D. positive value or zero

For 0 < x < 1, ln x is

A. positive

B. zero

C. negative

D. between 0 and 1

ANSWER: C. negative

If 1 < N < 10, then

A. 1 < log N < 2

B. 0 < log N < 1

C. 2 < log N < 3

D. -1 < log N < 0

ANSWER: A. 1 < log N < 2

To change logₐ to log𝘣N, multiply logₐN by

A. logₐb

B. log𝘣a

C. logɴa

D. logɴb

ANSWER: B. log𝘣a

The numbers logₐb and log𝘣a are

A. equal

B. equal but different in signs

C. reciprocal to each other

D. negative reciprocal to each other

ANSWER: C. reciprocal to each other

Logarithm using 10 as base.

A. decimal logarithm

B. scientific logarithm

C. common logarithm

D. natural logarithm

ANSWER: C. common logarithm

The logarithm of a product is the _____ of the logarithms, and the logarithm of a quotient is the _____ of the logarithms.

A. sum, difference

B. difference, sum

C. quotient, product

D. product, quotient

ANSWER: A. sum, difference

When a logarithm is expressed as an integer plus a decimal (between 0 and 1), the integer is called

A. Briggsian logarithm

B. Naperian logarithm

C. mantissa

D. characteristic

ANSWER: D. characteristic

The characteristic of a logarithm is 3 if the number is between

A. 1 and 10

B. 10 and 100

C. 100 and 1000

D. 1000 and 10000

ANSWER: D. 1000 and 10000

The characteristics of a common logarithm of a number greater than 1 is

A. zero

B. positive

C. negative

D. zero or positive

ANSWER: D. zero or positive

The characteristic is _____ the exponent of 10, when the number is written in scientific notation.

A. equal to

B. greater than

C. less than

D. none of the above

ANSWER: A. equal to

If logarithm to base 10 (denoted as log₁₀) is called common logarithm, what do you call the logarithm of base 2 (denoted as lb)?

A. binary logarithm

B. bit logarithm

C. bilogarithm

D. all of the above

ANSWER: A. binary logarithm

If the unknown is a conditional equation occurs as an exponent, the best way to solve the unknown is by

A. raising the power of both sides

B. taking the logarithm of both sides

C. extracting the root of both sides

D. applying the Newton's method

ANSWER: B. taking the logarithm of both sides

Angles of rotation with the same initial side and terminal side.

A. terminal angles

B. conjugate angles

C. coterminal angles

D. supplementary angles

ANSWER: C. coterminal angles

An angle equal to one revolution of 360°.

A. perigon

B. experiment angle

C. reflex angle

D. supplement angle

ANSWER: A. perigon

The angle which the line of sight to the object makes with the horizontal is above the eye of an observer.

A. angle of depression

B. angle of elevation

C. acute angle

D. bearing

ANSWER: B. angle of elevation

The angle which the line of sight to the object makes with the horizontal is below the eye of an observer.

A. angle of depression

B. angle of elevation

C. acute angle

D. bearing

ANSWER: A. angle of depression

A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides from same point inside the given triangle.

A. inscribed triangle

B. primitive triangle

C. pedal triangle

D. obtuse triangle

ANSWER: C. pedal triangle

The triangle with minimum perimeter but maximum area inscribed in another triangle is known as

A. pedal triangle

B. Euclid's triangle

C. primitive triangle

D. none of the above

ANSWER: A. pedal triangle

A right triangle whose lengths of sides may be expressed as ratio of integral units.

A. pedal triangle

B. isosceles triangle

C. scalene triangle

D. primitive triangle

ANSWER: D. primitive triangle

A triangle with no side equal is known as

A. acute triangle

B. oblique triangle

C. equilateral triangle

D. scalene triangle

ANSWER: D. scalene triangle

If two triangles have congruent bases, then the ratio of their areas equals the ratio of

A. their perimeters

B. the lengths of their altitudes

C. their sides

D. none of the above

ANSWER: B. the lengths of their altitudes

In an isosceles right triangle, the hypotenuse is _____ times as long as each of the legs.

A. √2

B. √3

C. 2

D. 3

ANSWER: A. √2

Which of the following is not a secondary part of a triangle?

A. altitudes

B. medians

C. exterior angles

D. sides

ANSWER: D. sides

Which of the following is not a property of a triangle?

A. The sum of the three angles is always equal to two right angles.

B. The sum of two sides is less than the third side.

C. If the two sides are equal, the angles opposite are unequal.

D. The altitudes of the triangle meet in a point.

ANSWER: B. The sum of two sides is less than the third side.

Given the sides of a triangle as 3m and 5m, the third side is

A. between 3m and 8m

B. greater than 8m

C. from 3m to 7m

D. from 2m to 8m

ANSWER: C. from 3m to 7m

A straight line from the vertex of a triangle to the midpoint of the opposite side is known as

A. altitude

B. median

C. height

D. A or B

ANSWER: B. median

Indicate the FALSE statement.

A. An altitude of a triangle is a perpendicular drop from any vertex to the opposite side.

B. Three or more lines which have one point in common are said to be coplanar.

C. The altitudes of a triangle meet in a point.

D. A locus is a figure containing all the points and only those which fulfill a given requirement.

ANSWER: B. Three or more lines which have one point in common are said to be coplanar.

The case of the solution of the triangle in the plane where the given data lead to two solutions.

A. axioms of Euclid

B. absurd case

C. ambiguous case

D. all of the above

ANSWER: C. ambiguous case

The most proved theorem in Mathematics.

A. Gauss lemma

B. Fermat's theorem

C. Ptolemy's theorem

D. Pythagorean theorem

ANSWER: D. Pythagorean theorem

The least proved theorem in Mathematics.

A. Goldbach conjecture

B. Fermat's last theorem

C. Mersenne's proportion

D. Pappus proportions

ANSWER: B. Fermat's last theorem

Equations used for checking the solution to a plane triangle using law of sines are as follows: a+b/c = cos½ (A - B) / sin ½C and a-b/c = cos½ (A - B) / sin ½C. These equations are called

A. Diophantine equations

B. Mollweide's equations

C. Mohr equations

D. Gauss equations

ANSWER: B. Mollweide's equations

Napier's rule states that the sine of any middle part is equal to the product of the _____ of the opposite parts.

A. sine

B. cosine

C. tangent

D. secant

ANSWER: B. cosine

Napier's rule states that the sine of any middle part is equal to the product of the _____ of the adjacent parts.

A. sine

B. cosine

C. tangent

D. secant

ANSWER: C. tangent

How many formulas may be derived from using the Napier's rule?

A. 5

B. 6

C. 8

D. 10

ANSWER: D. 10

The sum of all interior angles in a spherical triangle is always

A. greater than 180° but less than 270°

B. greater than 180° but less than 360°

C. greater than 180° but less than 540°

D. greater than 270° but less than 540°

ANSWER: C. greater than 180° but less than 540°

The maximum value for the longitude is

A. 90°

B. 180°

C. 45°

D. 360°

ANSWER: B. 180°

The maximum value for the latitude is

A. 90°

B. 45°

C. 180°

D. 360°

ANSWER: A. 90°

If R is the radius of a sphere and E is the spherical excess (in radians), then the area of a spherical triangle is

A. πR²E

B. R²E

C. ½R²E

D. R²/E

ANSWER: B. R²E

One minute of the great circle arc on the surface of the earth is equivalent to

A. 1 statute mile

B. 1 nautical mile

C. 60 statute mile

D. 60 nautical mile

ANSWER: B. 1 nautical mile

A spherical triangle with all angles equals to a right triangle is called _____ spherical triangle.

A. birectangular

B. quadrantal

C. trirectangular

D. right

ANSWER: A. birectangular

A spherical triangle with at least one side is a quarter of a great circle is called _____ spherical triangle.

A. birectangular

B. quadrantal

C. trirectangular

D. right

ANSWER: C. trirectangular

One of the two great circles intersecting at right angle at the piles and dividing equinoctial points and ecliptic into 4 parts.

A. nadir

B. zenith

C. declination

D. colure

ANSWER: D. colure

The radius of the earth used in spherical trigonometry is

A. 3989 statute miles

B. 3979 statute miles

C. 3969 statute miles

D. 3959 statute miles

ANSWER: D. 3959 statute miles

The difference between a nautical mile and a statute mile.

A. 800 feet

B. 900 feet

C. 1000 feet

D. 500 feet

ANSWER: A. 800 feet

Manila has a longitude of 121°05'E. What is the time difference between Manila and Greenwich, England which is at prime meridian?

A. 8 hours and 40 minutes

B. 8 hours and 34 minutes

C. 8 hours and 14 minutes

D. 8 hours and 4 minutes

ANSWER: D. 8 hours and 4 minutes

The earth is divided into how many time zones?

A. 24

B. 18

C. 16

D. 12

ANSWER: A. 24

In a spherical triangle, two angles (or sides) are on the same species if they are both

A. between 0° and 180°

B. between 0° and 90°

C. between 90° and 180°

D. between 0° and 90° or both between 90° and 180°

ANSWER: D. between 0° and 90° or both between 90° and 180°

Spherical degree is a unit of a spherical area taken as 1/720 of the surface of the sphere. How many spherical degrees does a hemisphere have?

A. 360°

B. 720°

C. 180°

D. 270°

ANSWER: A. 360°

Which of the following statements is FALSE about spherical trigonometry?

A. If two angles of a spherical triangle are equal, the sides opposite are equal, and conversely.

B. If two angles of a spherical triangle are unequal, and the greater side lies opposite the greater angle, and conversely.

C. The sum of two sides of a spherical triangle is greater than the third side.

D. The sum of all interior angles of a spherical triangle is 360°

ANSWER: D. The sum of all interior angles of a spherical triangle is 360°

The sum of the sides of a spherical triangle is always less than

A. 270°

B. 360°

C. 540°

D. 180°

ANSWER: B. 360°

The sum of any two angles of a spherical triangle is...

A. greater than 180°

B. less than 180°

C. less than 180° + the third angle

D. greater than 180° + the third angle Slowwor: C

ANSWER: C. less than 180° + the third angle

Refers to the angular distance from the equator measured along a meridian.

A. longitude

B. latitude

C. meridian

D. declination

ANSWER: B. latitude

Refers to the angle at either pole between the meridian passing through a point and some fixed meridian known as the prime meridian.

A. longitude

B. latitude

C. declination

D. equinox

ANSWER: A. longitude

Is half of a great circle terminated by the North Pole and South Pole.

A. longitude

B. latitude

C. declination

D. meridian

ANSWER: D. meridian

When the hypotenuse of a right spherical triangle is less than 90°.

A. the two legs are on the same quadrant

B. the two legs are on different quadrant

C. one leg is on the first quadrant and the other on the second quadrant

D. none of the above

ANSWER: A. the two legs are on the same quadrant

When the hypotenuse of a right spherical triangle is greater than 90°.

A. the two legs are on the same quadrant

B. the two legs are on different quadrant

C. one leg is on the first quadrant and the other on the second quadrant

D. none of the above

ANSWER: C. one leg is on the first quadrant and the other on the second quadrant

The point where a ray from the center of the earth through an observer's position on it intersects the celestial sphere is called the observer's

A. zenith

B. nadir

C. pole

D. equinox

ANSWER: A. zenith

The point that is diametrically opposite the zenith is called

A. pole

B. equinox

C. nadir

D. celestial meridian

ANSWER: C. nadir

The great circles through the north and south celestial poles are called

A. hour circles

B. celestial meridians

C. elevated poles

D. A and B

ANSWER: D. A and B

An oblique equilateral parallelogram.

A. square

B. rectangle

C. rhombus

D. trapezoid

ANSWER: C. rhombus

Mil is a unit of

A. angle

B. length

C. angle and length

D. weight

ANSWER: C. angle and length