Precalculus
0Loading crowd target
0Loading crowd target
6correct to beat the crowd
50 free Precalculus trivia questions with answers — science & nature quiz, new questions added Oct 2026.
Precalculus is the course that gathers up everything calculus will lean on: functions and their graphs, trigonometry, exponents and logarithms, sequences, vectors and the curves you get by slicing a cone. It is also where school math starts to explain the real world, from planetary orbits and compound interest to sound waves and navigation. This quiz mixes quick practice problems with the stories behind the ideas. You will convert between degrees and radians, find coterminal angles, continue arithmetic and geometric sequences, add vectors and read a graph for what it is telling you, then meet the mathematicians who invented the notation and the mnemonics students still use. It suits beginners meeting the subject for the first time, students reviewing before a test and teachers who want a ready-made warm-up. Every question comes with its answer and a short explanation, so a wrong guess still teaches something.
30 of 50 questions with answers and explanations. Play the quiz
Q 01Young Carl Friedrich Gauss is said to have quickly summed the whole numbers 1 to 100; what total did he reach?
5,050
5,050 is the total: pairing 1 with 100, 2 with 99 and so on gives 50 pairs that each add up to 101. The same trick sums any arithmetic series.
Q 02The mnemonic All Students Take Calculus helps trigonometry students recall which fact about the six trig functions?
Their signs in each quadrant
Their signs in each quadrant are what the initials track: all are positive in the first quadrant, only sine in the second, tangent in the third and cosine in the fourth.
Q 03Johannes Kepler's first law says each planet travels around the Sun along which curve rather than a perfect circle?
Ellipse
An ellipse, a stretched circle, is the path; Kepler published the law in 1609 after fitting Tycho Brahe's careful observations of Mars.
Q 04Which French philosopher, living in the Netherlands at the time, gives the Cartesian coordinate plane its name?
René Descartes
René Descartes set out the idea in 1637, letting algebra describe geometry; Pierre de Fermat found it independently but never published.
Q 05One radian is the angle at a circle's center whose arc is exactly as long as which part of the circle?
Radius
The radius sets the size of the unit, so one radian is the same angle on a coin as on a planet's orbit: about 57.3 degrees.
Q 06Which famous list, starting 0, 1, 1, 2, 3, 5, 8, makes each new term the sum of the previous two?
Fibonacci sequence
The Fibonacci sequence reached Europe in a 1202 book by Leonardo of Pisa, who used it to model a growing population of rabbits.
Q 07The identity sin²θ + cos²θ = 1 is named after which ancient Greek mathematician?
Pythagoras
Pythagoras gets the credit because the identity is his theorem in disguise: sin θ and cos θ are the two legs of a right triangle whose longest side measures 1.
Q 08Which capital Greek letter signals adding up the terms of a series in mathematical notation?
Sigma
Sigma stands for sum; capital pi does the same job for multiplication, signaling the product of a run of terms.
Q 09Jacob Bernoulli met the constant e while studying compound interest; e is approximately equal to which value?
2.718
2.718 is e to three decimal places: it is what $1 grows to in a year at 100% interest when the interest is compounded continuously.
Q 10A full 360° turn around a circle equals which angle in radians, roughly 6.28?
2π
2π radians make a full turn, because a circle's circumference is 2π times as long as the line from its center to its edge.
Q 11Which angle is coterminal with 45°, ending on the same terminal side after one extra full turn?
405°
405° is 45° plus one 360° rotation, so both angles point the same way; subtracting a turn instead gives −315°, which is coterminal too.
Q 12Which positive angle is coterminal with −60°, found by adding one full 360° turn to the negative angle?
300°
300° lands on the same terminal side as −60° because the two differ by exactly one full turn: 60° clockwise and 300° counterclockwise finish in the same place.
Q 13Which trigonometric function is the reciprocal of sine, so it is undefined wherever sin θ equals zero?
Cosecant
Cosecant is 1 over sine, a pairing that trips students up because the co-function goes with sine while secant goes with cosine.
Q 21Each term of the geometric sequence 2, 6, 18, 54 is three times the one before; which term comes next?
162
162 follows, three times 54. Growth by a fixed multiplier soon outruns growth by a fixed step, which is why this pattern describes compound interest.
Q 22The factorial 5!, the product of every whole number from 5 down to 1, equals which number?
120
120 is 5 × 4 × 3 × 2 × 1, and it is also the number of different orders in which five books can be lined up on a shelf.
Q 23Zeno's walker covers half the distance, then a quarter, then an eighth, forever; those fractions add up to which number?
1
1 is the total, which answers Zeno: the walker's endless list of ever-shorter steps still covers a finite distance.
Q 14What is the exact value of sin 30°, the shortest side over the longest in a 30-60-90 triangle?
1/2
1/2 is the value because a 30-60-90 triangle is half of an equilateral triangle, so its shortest side is half its longest.
Q 15The graph of y = tan x repeats its pattern after which interval, half as long as the sine wave's cycle?
180°
180° is the period of tangent, because opposite points on a circle give the same slope; sine and cosine need a full 360° before they repeat.
Q 16In y = 3 sin x, the 3 sets the wave's peak height above the x-axis; what is that height called?
Amplitude
Amplitude is the peak height of a wave; in a sound wave it relates to the volume, while frequency sets the pitch.
Q 17Which name is given to the curve x² + y² = 1, whose points are (cos θ, sin θ)?
Unit circle
The unit circle lets sine and cosine take values for any angle, not just the acute angles that fit inside a right triangle.
Q 18Which law extends the right-triangle rule a² + b² = c² to any triangle, using two sides and the included angle?
Law of cosines
The law of cosines adds a correction term, −2ab cos C, to the familiar formula; when C is 90° that term vanishes and the right-triangle rule returns.
Q 19Which kind of angle, the acute angle between a terminal side and the x-axis, helps find trig values in any quadrant?
Reference
Reference angles let one small table do all the work: 150°, 210° and 330° all share the reference angle 30°, so their trig values match up to sign.
Q 20In the arithmetic sequence 5, 7, 9, 11, 13, the constant 2 added at every step is called what?
Common difference
The common difference can be negative too: 10, 7, 4, 1 is still an arithmetic sequence, with a common difference of −3.
Q 24Which number array, named in the West after a French mathematician, lists binomial coefficients row by row?
Pascal's triangle
Pascal's triangle was studied centuries before him in India, Persia and China; in Iran it is known as Khayyam's triangle, after Omar Khayyam.
Q 25Expanding (x + y)² gives x² plus y² plus which middle term, the one that beginners often leave out?
2xy
2xy is the term; writing (x + y)² = x² + y² without it is such a common slip that it is nicknamed the freshman's dream.
Q 26Which series adds the reciprocals of the counting numbers and grows without bound though its terms shrink toward zero?
Harmonic
The harmonic series eventually passes every number, a fact Nicole Oresme proved in the 14th century, long before calculus existed.
Q 27Adding the first few odd numbers, as in 1 + 3 + 5 + 7, always produces which kind of total?
Perfect square
A perfect square appears every time: 1 + 3 + 5 + 7 = 16, because each new odd number wraps an L-shaped border around the square before it.
Q 28A vector runs 3 units right and 4 units up, written 3i + 4j; which magnitude, or arrow length, does that give?
5
5 is the length, since 3² + 4² = 25 and the square root of 25 is 5; the 3-4-5 right triangle is the best-known one with whole-number sides.
Q 29Speed and temperature have a size but no direction; what is such a quantity called, in contrast to a vector?
Scalar
A scalar is fully described by one number, which is why speed is a scalar while velocity, which also needs a direction, is a vector.
Q 30Two nonzero vectors that meet at a right angle, such as i and j, always have which dot product?
0
0 is the dot product of any perpendicular pair, which gives a quick test for right angles: multiply matching components, add, and see if the total vanishes.