2.4.4 · D5Trigonometry — Foundation
Question bank — Trig ratios of standard angles — 0°, 30°, 45°, 60°, 90° (derive, don't memorize blindly)
This is a thinking gym, not a calculator drill. Every item below hides a common trap — a place where the picture in your head goes wrong. Read the question, cover the answer, and say your reasoning out loud before revealing. If you cannot justify it in one breath, you have found a gap worth filling from the parent note the main topic.
True or false — justify
because 30° and 60° are the same size
False reason — they are equal, but not because the angles are equal. It is because they are complementary (add to 90°), so sine of one equals cosine of the other. See Complementary Angles.
tells us the opposite and adjacent sides of a 45° right triangle are equal
True — , and in the isosceles right triangle both legs are length 1, so the ratio is exactly 1.
can be larger than 1 for a steep enough angle
False — on the Unit Circle is a height inside a radius-1 circle, so it can never exceed 1; its maximum is exactly 1 at 90°.
means at 0° the whole hypotenuse lies along the horizontal
True — at 0° the triangle flattens onto the x-axis, so the adjacent side equals the hypotenuse and their ratio is 1.
equals a very large number
False — it is undefined, not "large". At 90° the adjacent side is 0, so , which is division by zero, not a number.
and are two different values for
False — they are the same number written differently; multiplying top and bottom by (Rationalization) turns one into the other.
As the angle grows from 0° to 90°, decreases the whole way
True — the adjacent (horizontal) side shrinks steadily from full length to 0, so falls from 1 to 0.
only works for 30°
False — this is the Pythagorean identity, true for every angle, since it just restates Pythagorean Theorem on a radius-1 triangle.
The 30-60-90 triangle must have sides exactly
False — those are the ratios; any similar triangle scales them (e.g. ). The trig ratios stay identical because ratios ignore size.
Spot the error
" because 60° is bigger, so its side is bigger"
The conclusion contradicts the premise — a bigger angle does have a bigger opposite side, so , not . The value belongs to 30°.
" because over "
Wrong final value — , not . The 2's cancel; you cannot keep one.
"Hypotenuse of the 45° triangle is since "
The error is skipping the square root — is the hypotenuse squared, so the hypotenuse is , not 2.
" because the point is at the very top of the circle"
At the top the point is , so the horizontal coordinate is 0 — that means . The value 1 there is .
", undefined"
Flipped it — at 0° opposite is 0 and adjacent is 1, so . The undefined case is 90°, where the denominator is 0.
", so "
Cosine is not the reciprocal of sine. For 45° both legs are equal, so ; the reciprocal is impossible for a cosine.
"For a ladder problem I used to link the vertical height to the ladder"
Cosine connects the adjacent (horizontal) side to the hypotenuse. A vertical height is the opposite side, so you need .
Why questions
Why do we rationalize into ?
To keep radicals out of denominators, which makes adding and comparing fractions far cleaner — e.g. . See Rationalization.
Why can we derive these values instead of memorizing them?
Because they come from two fixed shapes — the isosceles right triangle and the split equilateral triangle — plus Pythagorean Theorem. A 20-second sketch regenerates the whole table.
Why does dropping a perpendicular in an equilateral triangle bisect the base?
Because the equilateral triangle has mirror symmetry about that line, so the two halves are congruent and the base splits into two equal 1's. See Symmetry in Trigonometry.
Why is undefined while ?
. At 90° the denominator (divide by zero), but at 0° the numerator over a nonzero denominator gives a clean 0.
Why do sine and cosine "swap" between an angle and its complement?
In a right triangle the two non-right angles share the same two legs, but what is "opposite" for one is "adjacent" for the other — so their sine/cosine roles trade. See Complementary Angles.
Why prefer exact values like over the decimal 13.86?
The exact form loses no information and stays correct through further algebra; the decimal is rounded and errors can accumulate. See Exact vs Approximate Values.
Why does the "0-1-2-3-4 under the root" sine pattern work?
It is a memory scaffold matching for ; it happens to reproduce the true derived values but is a mnemonic, not a proof.
Edge cases
At 0°, is the "triangle" still a triangle?
No — it degenerates into a flat line segment along the x-axis (zero height), which is exactly why . We treat 0° as a limiting case.
Does ever go negative among the five standard angles –?
No — from 0° to 90° the adjacent side is never negative, so cosine stays between 0 and 1. Negative cosines appear only past 90° (other quadrants on the Unit Circle).
Is there an angle in – where ?
Yes, exactly 45°, where both equal — the point where the two legs of the triangle are equal and .
What is the smallest angle whose tangent is undefined in this range?
90° is the only one — it is the boundary where the adjacent side collapses to zero and the ratio has no value.
Between 30° and 60°, which has the larger tangent, and why?
60°, since versus — a steeper angle climbs more per step forward.
If a triangle's legs are scaled from to , do the trig ratios change?
No — trig ratios depend only on the angle, and scaling keeps the triangle similar, so every ratio is unchanged.
Recall One-sentence self-test
If someone hands you a blank table of 0°/30°/45°/60°/90°, you should be able to draw two triangles, apply Pythagorean Theorem, and fill in all fifteen entries — no rote memory required.
Numbers I claim to be true above
, , , all verified below.