3.1.4 · D5Advanced Trigonometry
Question bank — Trig functions for angles beyond 90° — ASTC rule (All, Sin, Tan, Cos)
Everything here rests on the parent idea: for any angle, . See the parent note, and lean on Unit Circle, Reference Angles, Trigonometric Identities, Radians and Degrees, and Graphs of Sin Cos Tan as needed.
True or false — justify
TF1. "ASTC tells you how big a trig ratio is."
False — ASTC decides only the sign; the magnitude always comes from the ratio of the reference angle .
TF2. "In Quadrant III both sine and cosine are negative, yet is positive there."
True — , and a negative divided by a negative is positive, so in QIII even though sin and cos are both negative.
TF3. " is negative."
True — is in QIII where , and , so cosine is negative.
TF4. " and can never both be negative for the same ."
False — in QIII (–) the point has and , so both are negative simultaneously.
TF5. "The reference angle of is because you measure from the -axis."
False — the reference angle is , always measured to the nearest -axis, never the -axis.
TF6. " equals its reference-angle sine, and ASTC applies cleanly at ."
False — sits ON the axis, not inside a quadrant, so ASTC (a per-quadrant rule) doesn't classify it; here directly, .
TF7. "Adding to an angle never changes any of its trig ratios."
True — one full turn returns to the same point on the Unit Circle, so all ratios are -periodic (period for sin/cos, for tan).
TF8. "A negative angle must have a negative sine."
False — sign depends on the landing quadrant, not on the input's sign; e.g. lands in QIII where sine is negative, but lands in QI where sine is positive.
Spot the error
SE1. " since the reference angle is ."
Reference angle is right, but is in QIII where only tan is positive, so sine is negative: .
SE2. " because is even."
Evenness () is irrelevant here — is in QII where cosine is negative, so .
SE3. "For I keep sine and just fix the sign, so it equals in QII."
A shift forces a cofunction swap: . ASTC's "same ratio" shortcut only works for and shifts, not or .
SE4. "The reference angle of is ."
After removing the full turn you get , but that is still not acute — is in QII, so the reference angle is .
SE5. " is a big positive number, so tan is positive at ."
is undefined, not "big positive"; there's no value to assign a sign to, so ASTC simply doesn't apply on the axis.
SE6. " because the angle is negative."
lands in QIV where cosine is positive, so ; the minus sign of the angle does not automatically minus the ratio.
SE7. "Since QII reference angle is , the reference angle of is and of is , so bigger gives bigger reference."
The verdict on each value is correct but the pattern claim is wrong: as grows toward , the reference angle shrinks toward .
Why questions
WHY1. "Why does the sign of every ratio depend only on the signs of and ?"
Because the Unit Circle definitions ARE , , — the ratios are literally the coordinates (or their quotient), so their signs are exactly the coordinate signs.
WHY2. "Why is the reference angle measured to the -axis and not the -axis?"
So that it matches the acute triangle formed with the horizontal, whose sin/cos equal the same-named ratios of ; measuring to the -axis would swap sine and cosine and break the clean rule.
WHY3. "Why does the mnemonic go A, S, T, C in that specific order?"
Reading the positive-ratio column anticlockwise from QI gives All → Sin → Tan → Cos; it is derived from the coordinate signs, not chosen arbitrarily.
WHY4. "Why does tan have period while sin and cos have period ?"
Rotating by negates both and , so (ratio of two negated values) is unchanged, whereas sin and cos individually flip and only recover after a full .
WHY5. "Why can we subtract freely from a large angle before applying ASTC?"
Each full turn returns the terminal ray to the identical point , so the ratios are unchanged — subtracting just relocates the angle into without altering any value. See Radians and Degrees for the version.
WHY6. "Why does a shift swap the function but a shift doesn't?"
A turn maps the horizontal leg to the vertical leg (opposite ↔ adjacent), swapping sin and cos; a turn keeps the ray along the same line, so the function name stays and only the sign can change. See Trigonometric Identities.
WHY7. "Why must a reference angle satisfy ?"
It is defined as the acute gap between the terminal ray and the nearest half of the -axis; the ray can never be more than from the closest horizontal direction.
Edge cases
EC1. "What quadrant and sign does give for cosine?"
lies ON the positive -axis (no quadrant); ASTC doesn't classify it, but directly , .
EC2. "Which trig ratio is undefined at , and why?"
is actually defined (); the undefined tangents occur at and where makes divide by zero.
EC3. "A terminal ray lands exactly on the negative -axis (). What is the reference angle?"
— a boundary reference angle; here , , and is undefined ().
EC4. "Is the same as for trig purposes?"
Yes — one full clockwise turn returns to , so every ratio matches those of (, , ).
EC5. "For , both and equal and ; does ASTC add anything?"
In QI ASTC says "All positive," which is consistent but adds no correction — ASTC only earns its keep once you leave QI where signs start flipping.
EC6. "At the reference angle would be , an allowed value; so why is ASTC still inapplicable?"
The reference angle can be , but the point sits on the -axis between quadrants, so there's no single quadrant to read a sign from — evaluate , directly instead.
EC7. "If , what does that tell us about the quadrant?"
forces , so the ray is on the -axis ( or ) — a boundary, not a quadrant interior.
Connections
- Unit Circle — every sign here is just a coordinate sign.
- Reference Angles — supplies the magnitude these traps keep confusing with the sign.
- Trigonometric Identities — the / cofunction swaps behind SE3 and WHY6.
- Radians and Degrees — periodicity restated with .
- Graphs of Sin Cos Tan — sign per quadrant = above/below the axis on the curve.