Trigonometry
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50 free Trigonometry trivia questions with answers — science & nature quiz, new questions added Oct 2026.
Trigonometry began as a tool for astronomers who wanted to turn angles in the sky into distances, and it now sits behind map-making, navigation, engineering and the study of sound and light. At its heart is one idea: in a right triangle, the angles fix the ratios of the sides. This quiz starts with the basics, including SOH-CAH-TOA, the special angles and the values every student memorises, then moves through radians, the graphs of sine, cosine and tangent, the main identities and the rules used to solve any triangle. A final stretch covers the history: the Greek, Indian, Persian and European scholars who named the functions and compiled the first tables. It suits students revising for a test, teachers who want a ready-made class quiz, and anyone curious about how much school maths has stuck. Every question comes with its answer and a short explanation.
30 of 50 questions with answers and explanations. Play the quiz
Q 01In the trig mnemonic SOH-CAH-TOA, the letter O appears twice; which word does it stand for both times?
Opposite
Opposite is the O in both SOH and TOA: sine is opposite over hypotenuse and tangent is opposite over adjacent. The mnemonic is often sounded out as one word, a little like the volcano Krakatoa.
Q 02Mount Everest is named after George Everest, who directed the trigonometric mapping of India in which post?
Surveyor General
Surveyor General of India was his title, yet George Everest never saw the peak. His successor Andrew Waugh proposed the name after years of triangle calculations showed it to be the highest known.
Q 03The tangent function takes its name from the Latin word tangens, which describes a line doing what to a circle?
Touching
Touching is the meaning: a tangent line just grazes a circle at a single point. The secant, from the Latin secans for 'cutting', is the line that slices straight through it.
Q 04The degrees used in trigonometry are usually traced to Babylonian astronomers, who counted in which number system?
Base 60
Base 60 counting, known as sexagesimal, is also why one degree splits into 60 arcminutes and each arcminute into 60 arcseconds.
Q 05Which Greek letter, the eighth in its alphabet, is the usual symbol for an unknown angle in trigonometry?
Theta
Theta, written θ, stands for the unknown angle in formulas such as sin θ. Although it is the eighth letter, in the Greek numeral system it carried the value 9.
Q 06The law of cosines works for any triangle, but with a 90° angle it reduces to which famous formula?
Pythagorean theorem
The Pythagorean theorem is the special case: cos 90° is zero, so the extra term −2ab cos C drops out and leaves c² = a² + b².
Q 07A flagpole casts a shadow exactly as long as the pole is tall; how high is the Sun above the horizon?
45°
45° is the answer because equal height and shadow form a right triangle with two matching sides, and tan 45° = 1. A longer shadow would mean a lower Sun.
Q 08Which technique fixes a distant point's position by measuring only angles to it from both ends of a known baseline?
Triangulation
Triangulation needs just one measured baseline, since every other distance follows from the angles. Trilateration, by contrast, works from measured distances.
Q 09On a sine wave such as a pure musical tone, what name is given to the peak height above the centre line?
Amplitude
Amplitude is the peak deviation from zero, so the curve y = 3 sin x climbs to 3 and dips to −3. A pure tone is a sine wave made of one single frequency.
Q 10Which Swiss mathematician's famous formula ties sine and cosine to the number e and the imaginary unit i?
Leonhard Euler
Leonhard Euler's formula states that e^(ix) = cos x + i sin x. First published in 1748, it shows that the trig functions and the exponential function are two faces of the same thing.
Q 11In height-and-distance problems, the angle between the horizontal and an upward line of sight is the angle of what?
Elevation
Elevation is measured upward from the horizontal, while the angle of depression is measured downward to something below. Either one pairs with the tangent ratio to turn a known distance into a height.
Q 12A right triangle has shorter sides of 3 and 4 units; how long is its hypotenuse, the side facing the right angle?
5
5 completes the famous 3-4-5 triangle, since 9 + 16 = 25. It is the only right-triangle shape whose three sides go up in equal steps.
Q 13Which word describes sine and cosine, used to model sound and light waves because their values repeat in cycles?
Periodic
Q 21Which French mathematician showed that any repeating signal can be built by adding simple sine and cosine waves?
Joseph Fourier
Joseph Fourier hit on the idea while working out how heat spreads through a metal plate, and presented his first results in 1807. His series now turn up across acoustics, optics and signal processing.
Q 22Which trig function is the reciprocal of sine, even though its name begins with the prefix 'co'?
Cosecant
Cosecant, written csc, is 1 divided by sine, whereas the plain secant is the reciprocal of cosine. Each 'co' name pairs with the basic function that lacks one.
Q 23'All Students Take Calculus' tells which trig functions are positive in each of the coordinate plane's what?
Quadrants
Periodic functions repeat after a fixed interval, and sine and cosine are among the simplest of them, which makes the pair the standard building blocks for studying anything that cycles.
Q 14Dividing sin A by cos A gives which other trig ratio of the angle A, which also measures a line's slope?
tan A
The ratio is tan A, because the shared hypotenuse cancels when sin A is divided by cos A. Rise over run for a line tilted at angle A is the same quotient.
Q 15What is the exact value of sin 30°, found by cutting an equilateral triangle in half?
1/2
1/2 is exact: halving an equilateral triangle with sides of 2 leaves a 30° angle facing a side of 1, while the longest side is still 2.
Q 16Squaring sin θ and cos θ for any angle θ and adding the two results always gives which number?
1
1 is the total every time, because sin θ and cos θ are the two shorter sides of a right triangle whose hypotenuse has length 1.
Q 17Trig ratios depend only on the angle because all right triangles sharing the same acute angle are what?
Similar
Similar triangles have the same shape at any size, so their sides stay in proportion. That is why sin 30° is identical for a triangle on a page and one spanning a valley.
Q 18A straight angle of 180° equals how many radians, the standard unit of angle in the SI system?
π
π radians make a half-turn, because one radian is the angle whose arc equals the radius, and half a circle's edge is π radii long.
Q 19Which ancient Greek astronomer compiled the first known table of chords and is often called the father of trigonometry?
Hipparchus
Hipparchus drew up his chord table around 140 BC to tackle problems in astronomy. He is even better known for discovering the precession of the equinoxes.
Q 20The word sine comes from the Latin sinus, picked by a 12th-century translator of Arabic texts; what did sinus mean?
A bay
A bay or fold was the Latin meaning, a long way from the original idea: the Sanskrit word behind the Arabic term meant a bowstring, picturing the chord stretched across an arc.
Quadrants are the four regions of the coordinate plane. Going anticlockwise from the top right, the initials stand for All, Sine, Tangent and Cosine, the functions that stay positive in each.
Q 24Cosine is an even function, with a graph that mirrors across the vertical axis, so cos(−B) always equals what?
cos B
The answer is cos B, since flipping the sign of an angle leaves its cosine unchanged. Sine is an odd function instead, so sin(−B) comes out as −sin B.
Q 25The graph of y = sin x is an endless wave; after how many degrees does its pattern first start again?
360°
360° is one full turn, which brings the angle back to where it began. The tangent function repeats twice as fast, every 180°.
Q 26A square has sides of length 1; how long is its diagonal, the hypotenuse of the two right triangles it creates?
√2
√2, about 1.414, follows from 1² + 1² = 2. The same ratio shapes every sheet of A4 paper, so that cutting one in half gives two sheets of the same proportions.
Q 27Besides degrees and radians, many calculators offer which third angle unit, dividing a right angle into 100 parts?
Gradians
Gradians, also called gons, split a full turn into 400 parts and are used mainly for land measurement in Europe. They are the G on a calculator's DRG key.
Q 28Which law of optics uses the sines of two angles to predict how light bends on entering water or glass?
Snell's law
Snell's law says the sines of the incoming and refracted angles keep a fixed ratio for any two materials. In optics it is the rule behind ray tracing, which works out where each beam of light will go.
Q 29What is the exact value of tan 60°, the ratio of the longer leg to the shorter leg in a 30-60-90 triangle?
√3
√3, about 1.73, compares the height of an equilateral triangle with sides of 2 to half its base. Flip the fraction over and you have tan 30° instead.
Q 30What is the value of cos 180°, the angle that points in exactly the reverse direction to 0°?
−1
−1 is the value, because a half-turn carries the point on the circle to its far left, at (−1, 0), and the cosine is simply that x-coordinate.