MCR3U Angles from 0° to 360° and the CAST Rule: Notes and Practice

SOH CAH TOA only works inside a right triangle, so angles between $0°$ and $90°$ . But real problems (and the unit circle) need trig ratios for ANY angle from $0°$ to $360°$ — and beyond, using negative angles. A robot arm rotating $250°$ , a wind direction of $300°$ , or the angle a Ferris wheel has turned all go past $90°$ . The trick is to relate every angle back to an acute 'reference angle' inside a right triangle, and track the sign separately. Two small ideas do all the work: a point's coordinates carry signs, and the distance to the origin never does. Master them and every angle, however large, reduces to one of the acute angles you already know.

Standard position and a point on the terminal arm

Standard position means an angle's initial arm lies along the positive $x$ -axis and its terminal arm rotates counterclockwise from there. Pick any point $P (x, y)$ on the terminal arm and let $r$ be its distance from the origin. Dropping a perpendicular from $P$ to the $x$ -axis makes a right triangle with legs $∣ x ∣$ and $∣ y ∣$ and hypotenuse $r$ , so the definitions below are just SOH CAH TOA with the coordinates keeping their signs. The ratios don't depend on which point you pick: a point twice as far out doubles $x$ , $y$ and $r$ together, so the ratios are unchanged. With $r = 1$ this is exactly the unit circle, where $x = cos θ$ and $y = sin θ$ .

r = x^{2} + y^{2} sin θ = \frac{y}{r} cos θ = \frac{x}{r} tan θ = \frac{y}{x}

Trig ratios of any angle from a point on its terminal arm. The distance r is always positive.

Quadrants and reference angles

The plane splits into four quadrants. The REFERENCE ANGLE (also called the related acute angle) is the acute angle between the terminal arm and the nearest $x$ -axis — it's always between $0°$ and $90°$ , and its trig ratios have the exact same size as the original angle's, just possibly a different sign. Why? Reflecting the terminal arm across an axis moves the point $P (x, y)$ to a spot with the same $∣ x ∣$ , $∣ y ∣$ and $r$ , so only the signs can change. The formula for the reference angle depends on how far the arm is from $180°$ or $360°$ , the two ends of the $x$ -axis.

Quadrant	$θ$ range	Reference angle formula	Signs of $x$ and $y$
I	$0°$ to $90°$	$ref = θ$	$x > 0, y > 0$
II	$90°$ to $180°$	$ref = 180° - θ$	$x < 0, y > 0$
III	$180°$ to $270°$	$ref = θ - 180°$	$x < 0, y < 0$
IV	$270°$ to $360°$	$ref = 360° - θ$	$x > 0, y < 0$

The CAST rule

Because $r$ is always positive, the sign of each ratio is decided entirely by the signs of $x$ and $y$ . Sine is $\frac{y}{r}$ , so it is positive wherever $y > 0$ (above the $x$ -axis, QI and QII). Cosine is $\frac{x}{r}$ , positive wherever $x > 0$ (right of the $y$ -axis, QI and QIV). Tangent is $\frac{y}{x}$ , positive when $x$ and $y$ share a sign (QI and QIII). Each quadrant therefore makes only certain ratios positive; the rest come out negative. Reading the quadrants counterclockwise starting from quadrant IV gives the mnemonic C-A-S-T.

Fig 1All ratios positive in QI, only Sine in QII, only Tangent in QIII, only Cosine in QIV.

Quadrant I — ALL three ratios (sin, cos, tan) are positive.
Quadrant II — only SINE (and its reciprocal, cosecant) is positive.
Quadrant III — only TANGENT (and its reciprocal, cotangent) is positive.
Quadrant IV — only COSINE (and its reciprocal, secant) is positive.

VideoAn Ontario MCR3U lesson on angles in standard position: the terminal arm, the related acute angle, and how the quadrant decides whether each ratio is positive or negative.Watch on YouTube

Negative angles, coterminal angles and reciprocal ratios

A negative angle rotates CLOCKWISE from the positive $x$ -axis instead of counterclockwise. $- 30°$ lands in the same place as $330°$ (add $360°$ to a negative angle to find its positive coterminal partner), so their trig ratios match exactly. Angles that share a terminal arm are coterminal, and they always differ by a multiple of $360°$ , so $390°$ also behaves exactly like $30°$ . For example, to evaluate $cos (- 120°)$ , add $360°$ to get the coterminal angle $240°$ . That is in QIII with reference angle $240° - 180° = 60°$ , and cosine is negative in QIII, so $cos (- 120°) = - cos 60° = - \frac{1}{2}$ . Notice the answer is negative because of the QUADRANT, not because the angle was negative: $cos (- 60°) = cos 300° = + \frac{1}{2}$ , since $300°$ is in QIV where cosine is positive. The reciprocal ratios flip sine, cosine, and tangent upside down: $csc θ = \frac{1}{sin θ} = \frac{r}{y}$ , $sec θ = \frac{1}{cos θ} = \frac{r}{x}$ , $cot θ = \frac{1}{tan θ} = \frac{x}{y}$ . A reciprocal ratio has the exact same sign as the ratio it comes from, because flipping a fraction never changes its sign.

Solving for the angle: expect two answers

Ignore the sign and find the reference angle from the positive value, e.g. for $sin θ = - 0.4$ use $sin^{- 1} (0.4) \approx 23.6°$ .
Use CAST and the sign to choose the two quadrants: sine negative means QIII and QIV.
Place the reference angle in each quadrant: QII $θ = 180° - ref$ , QIII $θ = 180° + ref$ , QIV $θ = 360° - ref$ . Here $180° + 23.6° = 203.6°$ and $360° - 23.6° = 336.4°$ .
Check each answer on the calculator: $sin 203.6° \approx - 0.40$ ✓ and $sin 336.4° \approx - 0.40$ ✓. A calculator's inverse key returns only ONE angle, which is why you must find the second one yourself.
A ratio of $0$ , or a sine or cosine of $1$ or $- 1$ , lands on an axis (a quadrantal angle such as $90°$ or $180°$ ) and gives only one or two answers, so read it straight off the unit circle.

Loading interactive…

Rotate the angle past 90, 180 and 270 degrees to see the reference angle and the sign flip in each quadrant.

Angles from 0° to 360° and the CAST Rule

Key things to remember

Standard position and a point on the terminal arm

Quadrants and reference angles

The CAST rule

Negative angles, coterminal angles and reciprocal ratios

Solving for the angle: expect two answers

Worked examples

Where marks get lost

Flashcards 9

Practice