Determine trigonometric ratios for angles from 0° to 360°

Ontario Grade 11 Mathematics

Trigonometric Functions

Using the Unit Circle and Reference Angles to Find Sine, Cosine, and Tangent in Every Quadrant

In Grade 10, you used sine, cosine, and tangent only for acute angles inside a right triangle. But angles in the real world — from the direction a force acts to the position of a point rotating around a centre — can be anywhere from 0° to 360°. This lesson extends the three primary trigonometric ratios to all such angles. You will see that right-triangle thinking still does most of the work; you only need a clear rule about signs to handle angles in every quadrant. No new ratios are introduced — just a broader stage for the familiar ones.

What you will learn

Explain what a reference angle is and how to find it for any angle from 0° to 360°.
Determine the sign of sine, cosine, and tangent in each of the four quadrants.
Calculate the exact or decimal value of a trigonometric ratio for any angle from 0° to 360°.
Recognise which quadrants produce a given positive or negative ratio value.

Prerequisite Bridge: Right-Triangle Trigonometry and the Cartesian Plane

Recall from Grade 10 that for an acute angle

\theta

inside a right triangle, the three primary ratios are defined as

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

,

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

, and

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

. These ratios are always positive because every side length is positive.

You also know the Cartesian plane: the horizontal axis is

x

and the vertical axis is

y

. The plane is divided into four quadrants. Quadrant I is top-right (both

x

and

y

positive), Quadrant II is top-left (

x

negative,

y

positive), Quadrant III is bottom-left (both negative), and Quadrant IV is bottom-right (

x

positive,

y

negative). Keeping this picture in mind is essential for the rest of the lesson.

SOH-CAH-TOA applies to acute angles in right triangles.
Side lengths are always positive; ratios are always positive for acute angles.
The four quadrants differ by the signs of the $x$ - and $y$ -coordinates.

Extending Trigonometric Ratios: The Rotating Arm Definition

To handle angles beyond 90°, imagine an arm of length

r > 0

anchored at the origin. The arm starts along the positive

x

-axis (pointing right) and rotates counter-clockwise. After rotating by angle

\theta

, the tip of the arm lands at some point

P = (x, y)

. The three ratios are now defined using those coordinates and the arm length

r

.

The new definitions are:

\sin\theta = \frac{y}{r}

,

\cos\theta = \frac{x}{r}

, and

\tan\theta = \frac{y}{x}

(provided

x \neq 0

). Notice that when

\theta

is acute and

P

is in Quadrant I, both

x

and

y

are positive, so these definitions match SOH-CAH-TOA exactly. The rotating-arm definition is simply an extension, not a replacement.

The arm length

r

is always positive because it is a distance:

r = \sqrt{x^2 + y^2}

. Because

r > 0

, the sign of

\sin\theta

depends entirely on the sign of

y

, and the sign of

\cos\theta

depends entirely on the sign of

x

. The sign of

\tan\theta

depends on whether

x

and

y

have the same sign or opposite signs.

\sin\theta = \frac{y}{r}, \cos\theta = \frac{x}{r}, \tan\theta = \frac{y}{x}

$\sin\theta = y/r$ , $\cos\theta = x/r$ , $\tan\theta = y/x$ .
$r$ is always positive; signs of the ratios come from the signs of $x$ and $y$ .
The definition matches Grade 10 SOH-CAH-TOA when $\theta$ is in Quadrant I.
$\tan\theta$ is undefined when $x = 0$ , i.e., at $\theta = 90°$ and $\theta = 270°$ .

Signs by Quadrant and the CAST Rule

Because

r

is always positive, the sign of each ratio in a given quadrant is completely determined by the signs of

x

and

y

there. In Quadrant I both

x > 0

and

y > 0

, so all three ratios are positive. In Quadrant II,

x < 0

and

y > 0

, so only sine is positive. In Quadrant III, both

x < 0

and

y < 0

, so only tangent is positive (a negative divided by a negative). In Quadrant IV,

x > 0

and

y < 0

, so only cosine is positive.

A popular memory device is the CAST rule. Reading the quadrants counter-clockwise starting from Quadrant IV, the letters C-A-S-T tell you which ratio is positive: Cosine (QIV), All (QI), Sine (QII), Tangent (QIII). You can also remember it clockwise from QI as 'All Students Take Calculus', but only use the initials A-S-T-C.

It is important to understand why the rule works, not just memorise it. If the tip of the arm is in Quadrant II, the

y

-coordinate is positive (above the

x

-axis) and the

x

-coordinate is negative (left of the

y

-axis). Dividing a positive

y

by a positive

r

gives a positive sine. Dividing a negative

x

by a positive

r

gives a negative cosine. Dividing a positive

y

by a negative

x

gives a negative tangent. The table below summarises all four quadrants.

In QI: $\sin > 0$ , $\cos > 0$ , $\tan > 0$ .
In QII: $\sin > 0$ , $\cos < 0$ , $\tan < 0$ .
In QIII: $\sin < 0$ , $\cos < 0$ , $\tan > 0$ .
In QIV: $\sin < 0$ , $\cos > 0$ , $\tan < 0$ .
CAST: starting at QIV counter-clockwise — Cosine, All, Sine, Tangent are positive.

Reference Angles: The Bridge Back to Acute Trigonometry

A reference angle is the acute angle (between 0° and 90°) formed between the arm and the nearest part of the

x

-axis. It is always positive and always less than or equal to 90°. The key insight is that the trigonometric ratio of any angle has the same absolute value as the ratio of its reference angle. You then attach the correct sign using the CAST rule.

Here is how to find the reference angle

\theta_R

for any angle

\theta

between 0° and 360°. If

\theta

is in Quadrant I (0° to 90°), then

\theta_R = \theta

. If

\theta

is in Quadrant II (90° to 180°), then

\theta_R = 180° - \theta

. If

\theta

is in Quadrant III (180° to 270°), then

\theta_R = \theta - 180°

. If

\theta

is in Quadrant IV (270° to 360°), then

\theta_R = 360° - \theta

.

For example, the reference angle for 210° is

210° - 180° = 30°

, and the reference angle for 315° is

360° - 315° = 45°

. Once you have the reference angle, you evaluate the ratio for that acute angle using a calculator or a known exact value, then apply the correct sign for the quadrant.

A reference angle is always between 0° and 90°.
QI: $\theta_R = \theta$ . QII: $\theta_R = 180° - \theta$ . QIII: $\theta_R = \theta - 180°$ . QIV: $\theta_R = 360° - \theta$ .
The ratio value equals the same ratio for $\theta_R$ , with the sign from the CAST rule.
Quadrantal angles (0°, 90°, 180°, 270°, 360°) are handled directly from the coordinates of the arm tip.

Quadrantal Angles: Special Cases at 0°, 90°, 180°, and 270°

When the rotating arm lands exactly on an axis, the angle is called a quadrantal angle. At these angles the tip of the arm sits on an axis, so one of

x

or

y

is zero. Using an arm of length

r = 1

for simplicity: at

\theta = 0°

, the tip is at

(1, 0)

, so

\sin 0° = 0

,

\cos 0° = 1

, and

\tan 0° = 0

. At

\theta = 90°

, the tip is at

(0, 1)

, so

\sin 90° = 1

,

\cos 90° = 0

, and

\tan 90°

is undefined (division by zero). At

\theta = 180°

, the tip is at

(-1, 0)

, so

\sin 180° = 0

,

\cos 180° = -1

, and

\tan 180° = 0

. At

\theta = 270°

, the tip is at

(0, -1)

, so

\sin 270° = -1

,

\cos 270° = 0

, and

\tan 270°

is undefined.

These values do not need a reference angle — read them directly from the coordinates. Memorising them saves time and helps you check your work on other angles.

Quadrantal angles sit on an axis; at least one of $x$ or $y$ is zero.
$\tan\theta$ is undefined whenever $x = 0$ , i.e., at 90° and 270°.
Use $r = 1$ to read the ratios directly: $\sin\theta = y$ , $\cos\theta = x$ .

Signs of Trigonometric Ratios by Quadrant (CAST Rule)

Quadrant	Angle Range	sin θ	cos θ	tan θ
I	0° to 90°	+ (positive)	+ (positive)	+ (positive)
II	90° to 180°	+ (positive)	− (negative)	− (negative)
III	180° to 270°	− (negative)	− (negative)	+ (positive)
IV	270° to 360°	− (negative)	+ (positive)	− (negative)

Worked example

Finding All Three Ratios for 150°

Determine the exact values of

\sin 150°

,

\cos 150°

, and

\tan 150°

.

Identify the quadrant
Since $90° < 150° < 180°$ , the angle is in Quadrant II. In QII, sine is positive, cosine is negative, and tangent is negative.
$150° ∈ \text{QII}$
Find the reference angle
For a Quadrant II angle, subtract from 180°: $\theta_R = 180° - 150° = 30°$ .
$\theta_R = 180° - 150° = 30°$
Write the ratios for the reference angle
From the special 30-60-90 triangle, we know $\sin 30° = \frac{1}{2}$ , $\cos 30° = \frac{\sqrt{3}}{2}$ , and $\tan 30° = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ .
$\sin 30° = \frac{1}{2}, \cos 30° = \frac{\sqrt{3}}{2}, \tan 30° = \frac{\sqrt{3}}{3}$
Apply the quadrant signs
In QII, sine keeps its positive sign, while cosine and tangent become negative. Copy the absolute value from the reference angle and attach the correct sign.
$\sin 150° = +\frac{1}{2}, \cos 150° = -\frac{\sqrt{3}}{2}, \tan 150° = -\frac{\sqrt{3}}{3}$
Verify using the ratio definition
Check that $\tan 150° = \frac{\sin 150°}{\cos 150°}$ . Dividing gives $\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{-2}{\sqrt{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$ . This matches, confirming the answer.
$\frac{\sin 150°}{\cos 150°} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Answer:

\sin 150° = \frac{1}{2}

,

\cos 150° = -\frac{\sqrt{3}}{2}

,

\tan 150° = -\frac{\sqrt{3}}{3}

Check: Using a calculator:

\sin 150° \approx 0.5000

✓,

\cos 150° \approx -0.8660 \approx -\frac{\sqrt{3}}{2}

✓,

\tan 150° \approx -0.5774 \approx -\frac{\sqrt{3}}{3}

✓.

Worked example

Finding All Three Ratios for an Angle Given a Point on the Terminal Arm

The terminal arm of angle

\theta

(in standard position) passes through the point

(-5, 12)

. Determine

\sin\theta

,

\cos\theta

, and

\tan\theta

, and state the quadrant in which

\theta

lies.

Identify the quadrant from the coordinates
The point is $(-5, 12)$ : the $x$ -coordinate is negative and the $y$ -coordinate is positive, so the point is in Quadrant II.
$x = -5, y = 12 \Rightarrow \text{QII}$
Calculate the arm length r
The arm length is the distance from the origin to $(-5, 12)$ . Apply the Pythagorean theorem: $r = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ .
$r = \sqrt{(-5)^2 + 12^2} = \sqrt{169} = 13$
Apply the rotating-arm definitions
Substitute $x = -5$ , $y = 12$ , and $r = 13$ into the three definitions: $\sin\theta = \frac{y}{r}$ , $\cos\theta = \frac{x}{r}$ , $\tan\theta = \frac{y}{x}$ .
$\sin\theta = \frac{12}{13}, \cos\theta = \frac{-5}{13}, \tan\theta = \frac{12}{-5}$
Simplify and check the signs
Write each ratio in simplest form. In QII, sine should be positive, cosine negative, and tangent negative — all three signs match the CAST rule, which confirms the calculations.
$\sin\theta = \frac{12}{13}, \cos\theta = -\frac{5}{13}, \tan\theta = -\frac{12}{5}$

Answer:

\sin\theta = \frac{12}{13}

,

\cos\theta = -\frac{5}{13}

,

\tan\theta = -\frac{12}{5}

;

\theta

is in Quadrant II.

Check: Verify:

\frac{\sin\theta}{\cos\theta} = \frac{\frac{12}{13}}{-\frac{5}{13}} = \frac{12}{13} \times \frac{-13}{5} = -\frac{12}{5}

, which equals

\tan\theta

✓. Also

r^2 = (-5)^2 + 12^2 = 25 + 144 = 169 = 13^2

✓.

Common mistakes and how to avoid them

Using the given angle directly in the calculator instead of the reference angle, and then ignoring the sign. For example, reporting

\sin 210° \approx -0.5

without explaining why it is negative.

Correction: Always find the reference angle first, then use the CAST rule to assign the correct sign. The calculator's result for

\sin 210°

is

-0.5

; confirm this equals

-(\sin 30°) = -\frac{1}{2}

because 210° is in QIII where sine is negative.

Thinking the reference angle formula is the same for every quadrant. A common error is using

180° - \theta

even when

\theta

is in Quadrant III or IV.

Correction: The formula depends on the quadrant: QII uses

180° - \theta

, QIII uses

\theta - 180°

, and QIV uses

360° - \theta

. Always identify the quadrant first.

Forgetting that

\tan\theta

is undefined at 90° and 270°, and entering these angles into a calculator expecting a number.

Correction: At 90° and 270°, the

x

-coordinate of the arm tip is 0. Since

\tan\theta = y/x

, division by zero is undefined. Recognise these as special cases and state that

\tan\theta

is undefined.

Confusing which single ratio is positive when applying the CAST rule. For example, thinking that in QIII, sine is positive because 'the angle is large'.

Correction: In QIII both

x < 0

and

y < 0

. Only

\tan\theta = y/x

is positive (negative ÷ negative).

\sin\theta = y/r

is negative and

\cos\theta = x/r

is negative. Always return to the coordinate signs.

When given a point on the terminal arm, forgetting to compute

r

and instead using one coordinate as the hypotenuse.

Correction: Always calculate

r = \sqrt{x^2 + y^2}

using both coordinates. The arm length

r

is the hypotenuse;

x

and

y

are the legs of the right triangle formed by dropping a perpendicular to the

x

-axis.

Lesson summary

The rotating-arm definition extends $\sin\theta = y/r$ , $\cos\theta = x/r$ , $\tan\theta = y/x$ to all angles from 0° to 360°, where $r = \sqrt{x^2+y^2} > 0$ .
The sign of each ratio in a quadrant depends on the signs of the $x$ - and $y$ -coordinates of the arm tip.
The CAST rule states which ratio is positive in each quadrant: Cosine in QIV, All in QI, Sine in QII, Tangent in QIII.
A reference angle is the acute angle between the terminal arm and the nearest part of the $x$ -axis; it is used to find the magnitude of the ratio.
Quadrantal angles (0°, 90°, 180°, 270°) are handled directly from the axis coordinates; $\tan\theta$ is undefined at 90° and 270°.
To evaluate any ratio: identify the quadrant, find the reference angle, evaluate for that acute angle, then apply the CAST sign.

Check your understanding

Question 1

What is the reference angle for

\theta = 250°

?

70°
80°
110°
250°

Show answer and explanation

70°

250° is in Quadrant III (between 180° and 270°). The reference angle for QIII is

\theta - 180° = 250° - 180° = 70°

.

Question 2

Which of the following correctly states the sign of

\cos 320°

?

Negative, because 320° is in Quadrant IV where cosine is negative.
Positive, because 320° is in Quadrant IV where cosine is positive.
Positive, because 320° is close to 360° and all ratios are positive near 360°.
Negative, because the reference angle is greater than 45°.

Show answer and explanation

Positive, because 320° is in Quadrant IV where cosine is positive.

320° is in Quadrant IV (between 270° and 360°). According to the CAST rule, cosine is positive in QIV because the

x

-coordinate is positive there.

Question 3

The terminal arm of angle

\theta

passes through

(−3, −4)

. What is

\sin\theta

?

$\frac{4}{5}$
$-\frac{3}{5}$
$-\frac{4}{5}$
$\frac{3}{4}$

Show answer and explanation

-\frac{4}{5}

First find

r = \sqrt{(-3)^2+(-4)^2} = \sqrt{9+16} = 5

. Then

\sin\theta = y/r = -4/5

. The point is in QIII where sine is negative, which confirms the sign.

Question 4

What is the exact value of

\tan 315°

?

$1$
$\frac{\sqrt{2}}{2}$
$-\frac{\sqrt{2}}{2}$
$-1$

Show answer and explanation

-1

315° is in Quadrant IV; its reference angle is

360° - 315° = 45°

. We know

\tan 45° = 1

. In QIV, tangent is negative, so

\tan 315° = -1

.

Key terms

Standard Position: An angle placed in the Cartesian plane with its vertex at the origin and its initial arm along the positive $x$ -axis. Rotation is counter-clockwise for positive angles.
Terminal Arm: The arm that rotates from the initial position to create the angle. The coordinates of a point on this arm are used to define the trigonometric ratios.
Reference Angle: The acute angle (from 0° to 90°) between the terminal arm and the nearest part of the $x$ -axis. It is always positive and is used to find the magnitude of a trigonometric ratio.
CAST Rule: A memory device indicating which primary trigonometric ratio is positive in each quadrant: Cosine (QIV), All (QI), Sine (QII), Tangent (QIII).
Quadrantal Angle: An angle whose terminal arm lies exactly on one of the coordinate axes: 0°, 90°, 180°, or 270°.
Arm Length (r): The distance from the origin to the tip of the terminal arm, calculated as $r = \sqrt{x^2 + y^2}$ . It is always positive.
Rotating-Arm Definition: The extension of sine, cosine, and tangent to all angles using a rotating arm: $\sin\theta = y/r$ , $\cos\theta = x/r$ , $\tan\theta = y/x$ , where $(x, y)$ is a point on the terminal arm and $r$ is the arm length.

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