Determine exact ratios for special radian angles

Ontario Grade 12 Mathematics

Trigonometric Functions

Use the unit circle, reference angles, and quadrant signs

An angle in radians may look less familiar than an angle in degrees, but special radian angles follow a small set of patterns. The unit circle connects each angle to a point whose coordinates give its cosine and sine. Once you know the first-quadrant values, the quadrant tells you the signs. This lesson focuses on exact ratios rather than decimal approximations.

What you will learn

Recall how radians and the unit circle describe an angle.
Determine exact sine, cosine, and tangent values for special radian angles.
Use a reference angle and quadrant signs to find ratios beyond the first quadrant.
Recognize when tangent is undefined.

1. Prerequisite bridge: radians and the unit circle

An angle measures a turn. Degrees and radians are two ways to describe that turn. A full turn is

360^\circ

or

2\pi

radians, so a half-turn is

180^\circ

or

\pi

radians. The symbol

\pi

is the number pi.

An angle in standard position starts on the positive horizontal axis and turns counterclockwise. Its terminal arm is the ray where the turn ends. A quadrant is one of the four regions of the coordinate plane.

The unit circle is a circle with radius

1

centred at the origin. For an angle

\theta

in standard position, the point where its terminal arm meets the unit circle has coordinates

(\cos\theta,\sin\theta)

. The horizontal coordinate is cosine, and the vertical coordinate is sine.

Tangent is the ratio of sine to cosine. It is undefined when cosine is

0

, because division by zero is not defined.

\tan\theta=\frac{\sin\theta}{\cos\theta}

One full turn is $2\pi$ radians.
On the unit circle, the point for $\theta$ is $(\cos\theta,\sin\theta)$ .
Tangent is sine divided by cosine.

2. First-quadrant special angles

The main first-quadrant special angles are

\frac{\pi}{6}

,

\frac{\pi}{4}

, and

\frac{\pi}{3}

. They correspond to

30^\circ

,

45^\circ

, and

60^\circ

. The angles

0

and

\frac{\pi}{2}

mark the horizontal and vertical ends of this quarter-turn.

For each angle, the unit-circle point gives cosine and sine together. For example, the point at

\frac{\pi}{3}

is

(\frac{1}{2},\frac{\sqrt{3}}{2})

. The first coordinate is cosine, and the second is sine.

The table lists the exact values. A radical, such as

\sqrt{2}

, is an exact value, not a rounded decimal. Find tangent by dividing sine by cosine. At

\frac{\pi}{4}

, sine and cosine are equal, so tangent is

1

.

(\cos\theta,\sin\theta)

Read cosine from the first coordinate and sine from the second.
Use the exact values in the table rather than rounded decimals.
Find tangent by dividing sine by cosine.

3. Other quadrants: reference angles and signs

A reference angle is the positive acute angle between an angle’s terminal arm and the horizontal axis. For example,

\frac{5\pi}{6}

is in Quadrant II, and its reference angle is

\frac{\pi}{6}

. The ratios have the same sizes as those at the reference angle, but their signs depend on the quadrant.

The coordinates show the signs. In Quadrant I, both are positive. In Quadrant II, the horizontal coordinate is negative and the vertical coordinate is positive. In Quadrant III, both are negative. In Quadrant IV, the horizontal coordinate is positive and the vertical coordinate is negative.

Cosine is the horizontal coordinate, and sine is the vertical coordinate. These coordinate signs determine their signs. Tangent is positive when sine and cosine have the same sign, and negative when they have opposite signs.

To find exact ratios for an angle in any quadrant, identify its reference angle first. Use the first-quadrant values for the exact size, then apply the sign for the angle’s quadrant. At a vertical-axis angle such as

\frac{\pi}{2}

, cosine is zero, so tangent is undefined.

\tan\theta=\frac{\sin\theta}{\cos\theta}

The reference angle gives the exact value sizes.
The quadrant gives the signs.
Tangent is undefined when cosine is zero.

4. A reliable method

When asked to determine exact ratios for a special radian angle, locate the angle on the unit circle and identify its quadrant. Find its reference angle and use the first-quadrant table to get the corresponding values. Apply the signs for the quadrant. If tangent is requested, divide sine by cosine, unless cosine is zero.

Check that each result is exact. Values such as

\frac{\sqrt{3}}{2}

and

-1

are exact. A decimal approximation alone does not give an exact ratio.

This process also helps catch sign errors. The sine value must match the vertical coordinate’s sign, and the cosine value must match the horizontal coordinate’s sign. The tangent sign must agree with the quotient of those signed values.

A useful sequence is: identify the angle, find its reference angle, then determine the exact value and sign.

Use the angle, reference angle, and quadrant in that order.
Check signs using the unit-circle coordinates.
Keep exact values rather than replacing them with decimals.

Exact ratios for first-quadrant special angles

Angle in radians	Angle in degrees	Unit-circle point	Sine	Cosine
$0$	$0^\circ$	$(1,0)$	$0$	$1$
$\frac{\pi}{6}$	$30^\circ$	$(\frac{\sqrt{3}}{2},\frac{1}{2})$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\frac{\pi}{4}$	$45^\circ$	$(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$\frac{\pi}{3}$	$60^\circ$	$(\frac{1}{2},\frac{\sqrt{3}}{2})$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{\pi}{2}$	$90^\circ$	$(0,1)$	$1$	$0$

Signs of sine, cosine, and tangent by quadrant

Quadrant	Sine	Cosine	Tangent
I	Positive	Positive	Positive
II	Positive	Negative	Negative
III	Negative	Negative	Positive
IV	Negative	Positive	Negative

Worked example

Find all three ratios for a Quadrant II angle

Determine the exact values of

\sin\frac{5\pi}{6}

,

\cos\frac{5\pi}{6}

, and

\tan\frac{5\pi}{6}

.

Locate the angle
The angle $\frac{5\pi}{6}$ is between $\frac{\pi}{2}$ and $\pi$ , so its terminal arm is in Quadrant II.
$\frac{\pi}{2}<\frac{5\pi}{6}<\pi$
Find the reference angle
In Quadrant II, subtract the angle from $\pi$ to get its reference angle. This gives $\frac{\pi}{6}$ , whose first-quadrant sine and cosine values are in the table.
$\pi-\frac{5\pi}{6}=\frac{\pi}{6}$
Apply the quadrant signs
At reference angle $\frac{\pi}{6}$ , sine is $\frac{1}{2}$ and cosine is $\frac{\sqrt{3}}{2}$ . In Quadrant II, sine is positive and cosine is negative.
$\sin\frac{5\pi}{6}=\frac{1}{2},\quad\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$
Find tangent
Divide sine by cosine. The result is negative because the numerator is positive and the denominator is negative.
$\tan\frac{5\pi}{6}=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=-\frac{\sqrt{3}}{3}$

Answer:

\sin\frac{5\pi}{6}=\frac{1}{2},\quad\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2},\quad\tan\frac{5\pi}{6}=-\frac{\sqrt{3}}{3}

Check: The sine is positive and cosine is negative in Quadrant II, so tangent must be negative. Dividing the two exact values gives

-\frac{\sqrt{3}}{3}

.

Common mistakes and how to avoid them

Using the reference-angle values but giving every ratio a positive sign.

Correction: Use the reference angle for the exact size, then use the quadrant to determine each sign.

Swapping sine and cosine.

Correction: On the unit circle, cosine is the horizontal coordinate and sine is the vertical coordinate.

Giving a tangent value when cosine is zero.

Correction: Tangent is sine divided by cosine. If cosine is zero, tangent is undefined.

Replacing an exact value with a rounded decimal.

Correction: Keep forms such as

\frac{\sqrt{2}}{2}

or

-\frac{\sqrt{3}}{3}

when an exact ratio is requested.

Lesson summary

The unit-circle point for $\theta$ is $(\cos\theta,\sin\theta)$ .
The first-quadrant table gives exact values for the special angles.
For other quadrants, use the reference angle for the value and the quadrant for the sign.
Tangent is the quotient of sine and cosine, and it is undefined when cosine is zero.

Check your understanding

Question 1

What is the exact value of

\cos\frac{2\pi}{3}

?

$\frac{1}{2}$
$-\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$-\frac{\sqrt{3}}{2}$

Show answer and explanation

-\frac{1}{2}

The reference angle is

\frac{\pi}{3}

, where cosine is

\frac{1}{2}

. The angle is in Quadrant II, where cosine is negative.

Question 2

What is the exact value of

\sin\frac{7\pi}{4}

?

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

Show answer and explanation

-\frac{\sqrt{2}}{2}

The reference angle is

\frac{\pi}{4}

, where sine is

\frac{\sqrt{2}}{2}

. The angle is in Quadrant IV, where sine is negative.

Question 3

What is the exact value of

\tan\frac{\pi}{3}

?

$\frac{\sqrt{3}}{3}$
$1$
$\sqrt{3}$
$-\sqrt{3}$

Show answer and explanation

\sqrt{3}

At

\frac{\pi}{3}

, sine is

\frac{\sqrt{3}}{2}

and cosine is

\frac{1}{2}

. Their quotient is

\sqrt{3}

.

Question 4

What is

\tan\frac{\pi}{2}

?

$0$
$1$
$-1$
Undefined

Show answer and explanation

Undefined

At

\frac{\pi}{2}

, cosine is

0

. Since tangent divides by cosine, the ratio is undefined.

Key terms

Radian: A unit for measuring angles. A full turn is $2\pi$ radians.
Unit circle: A circle centred at the origin with radius $1$ .
Terminal arm: The ray showing where an angle in standard position ends.
Reference angle: The positive acute angle between an angle’s terminal arm and the horizontal axis.
Exact value: A value written precisely, such as $\frac{\sqrt{3}}{2}$ , rather than as a rounded decimal.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B1.4. It is a study resource, not an official curriculum publication.

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