Use the unit circle and exact trigonometric values

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Geometry and Trigonometry

A visual and analytical guide for IB Mathematics: Analysis and Approaches SL, topic SL 3.5

This lesson focuses on using the unit circle and exact trigonometric values. You should already know that angles can be measured in degrees or radians and that a right-angled triangle has side ratios called sine, cosine, and tangent. On the unit circle, these ideas extend to angles in every direction around a circle. The circle gives a reliable way to identify exact values and their signs. Unless an angle is explicitly in degrees, angles in formulas in this lesson are in radians.

What you will learn

Explain how the unit circle represents sine and cosine.
Recall and use exact trigonometric values for standard angles.
Use a reference angle and quadrant signs to find exact values beyond the first quadrant.
Check angle locations and approximate values with a graphing calculator without replacing exact reasoning.

1. Prior knowledge: angles and side ratios

An angle describes a turn from a starting direction. One full turn is

360^\circ

or

2\pi

radians, so

180^\circ=\pi

radians. A positive angle is measured anticlockwise from the positive horizontal direction. For example,

\frac{\pi}{2}

radians is a quarter-turn.

In a right-angled triangle, sine is opposite side divided by hypotenuse, cosine is adjacent side divided by hypotenuse, and tangent is opposite side divided by adjacent side. The unit circle keeps the same sine and cosine relationships, but makes them usable for angles in all four quadrants.

180^\circ=\pi\text{ radians}

A full turn is $2\pi$ radians.
Degree and radian measures can describe the same angle.
Sine, cosine, and tangent are trigonometric ratios.

2. The unit circle and exact values

The unit circle is the circle of radius

1

centred at the origin of a coordinate plane. A point reached by turning through an angle

\theta

from the positive horizontal direction has coordinates

(\cos\theta,\sin\theta)

. Therefore, the horizontal coordinate gives cosine and the vertical coordinate gives sine. Tangent is the ratio of these coordinates when cosine is not zero.

The signs follow from the point's location: in the upper half of the circle sine is positive, and in the lower half it is negative. Cosine is positive on the right half and negative on the left. Tangent is positive when sine and cosine have the same sign, and negative when their signs differ.

The familiar exact values come from special right triangles. For angles in the first quadrant, the values at

\frac{\pi}{6}

,

\frac{\pi}{4}

, and

\frac{\pi}{3}

follow from triangles with side ratios

1:\sqrt{3}:2

and

1:1:\sqrt{2}

. At the axis angles, the coordinates are read directly from the circle. Knowing these values lets you give exact answers rather than rounded calculator decimals.

To use a standard value at an angle outside the first quadrant, find its reference angle: the acute angle between the terminal side and the horizontal axis. The reference angle supplies the size of the sine and cosine values; the quadrant supplies their signs. For instance, an angle in the second quadrant has the same reference-angle sine as its first-quadrant partner, but its cosine is negative.

(\cos\theta,\sin\theta)

A point on the unit circle at angle $\theta$ is $(\cos\theta,\sin\theta)$ .
Use the reference angle for the value and the quadrant for its sign.
Exact values use fractions and roots rather than rounded decimals.

3. Read values in symbolic, numerical, and graphical forms

A standard-angle table is a compact numerical record of the unit-circle coordinates. It is useful to learn the first-quadrant entries and then apply the quadrant signs. At

0

the point is

(1,0)

; at

\frac{\pi}{2}

it is

(0,1)

; at

\pi

it is

(-1,0)

; and at

\frac{3\pi}{2}

it is

(0,-1)

. These values also help you locate the axes before interpreting any other angle.

The graph of

y=\sin\theta

records the vertical coordinate as the angle changes, while

y=\cos\theta

records the horizontal coordinate. A graphing calculator can check an angle's approximate value or show whether a sign is plausible. Set the angle mode to radians when entering radian measures. The graph or decimal is a check: the unit-circle reasoning is what establishes an exact answer.

In a context, a point moving around a circle of radius

1

has horizontal and vertical coordinates given by cosine and sine. If the circle has radius

r

, the corresponding coordinates are

r\cos\theta

and

r\sin\theta

when its centre is at the origin. The radius and angle determine the position; the trigonometric values themselves remain dimensionless.

(x,y)=(r\cos\theta,r\sin\theta)

Check the calculator's angle mode before evaluating an angle.
Use an approximate decimal to check sign and size, not as a substitute for an exact value.
Coordinates in a circular context inherit the units of the radius.

4. A reliable method and technology check

For an exact-value question, first identify the angle's quadrant and, when useful, its reference angle. Match the reference angle to a standard value, then apply the correct sign. For a sum or difference of exact values, evaluate each trigonometric value before doing the arithmetic. Keep roots and fractions exact until the question asks for an approximation.

To check with technology, enter the angle in the correct mode and compare the calculator's decimal with the exact value's approximate size. A graph can also show whether the angle lies where the relevant function is positive or negative. For example, a second-quadrant angle should have positive sine and negative cosine. If the calculator gives the opposite signs, check the angle mode and the quadrant reasoning.

Angle location, reference angle, standard value, and sign form a dependable sequence.
Retain exact values unless a decimal is requested.
Technology checks calculations and interpretation; it does not explain why a value is exact.

Standard first-quadrant values and axis values

Angle $\theta$	$\cos\theta$	$\sin\theta$	$\tan\theta$
$0$	$1$	$0$	$0$
$\frac{\pi}{6}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{3}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$\frac{\pi}{3}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\sqrt{3}$
$\frac{\pi}{2}$	$0$	$1$	Undefined

Worked example

Find values in the second quadrant

Find the exact values of

\sin\left(\frac{5\pi}{6}\right)

and

\cos\left(\frac{5\pi}{6}\right)

.

Locate the angle
The angle $\frac{5\pi}{6}$ is between $\frac{\pi}{2}$ and $\pi$ , so its terminal side is in the second quadrant. Its reference angle is the difference between $\pi$ and the given angle.
$\pi-\frac{5\pi}{6}=\frac{\pi}{6}$
Use the standard values and signs
At the reference angle $\frac{\pi}{6}$ , sine is $\frac{1}{2}$ and cosine is $\frac{\sqrt{3}}{2}$ . In the second quadrant sine is positive and cosine is negative.
$\sin\left(\frac{5\pi}{6}\right)=\frac{1}{2},\quad \cos\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2}$

Answer:

\sin\left(\frac{5\pi}{6}\right)=\frac{1}{2}

and

\cos\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2}

.

Check: The point is in the upper-left part of the circle, so its vertical coordinate is positive and its horizontal coordinate is negative.

Worked example

Use a reference angle in the fourth quadrant

Find the exact value of

\tan\left(\frac{7\pi}{4}\right)

.

Identify the reference angle
The angle $\frac{7\pi}{4}$ lies in the fourth quadrant. Its reference angle is the difference between one full turn and the angle.
$2\pi-\frac{7\pi}{4}=\frac{\pi}{4}$
Apply the tangent value and sign
At $\frac{\pi}{4}$ , sine and cosine are both $\frac{\sqrt{2}}{2}$ , so tangent has magnitude $1$ . In the fourth quadrant sine is negative and cosine is positive, making tangent negative.
$\tan\left(\frac{7\pi}{4}\right)=-1$

Answer:

-1

Check: The coordinates have opposite signs, so their ratio, tangent, must be negative.

Worked example

Combine exact values

Evaluate

\sin\left(\frac{\pi}{3}\right)+\cos\left(\frac{\pi}{6}\right)

exactly.

Substitute standard values
Both angles are standard first-quadrant angles. The sine of $\frac{\pi}{3}$ and the cosine of $\frac{\pi}{6}$ are each $\frac{\sqrt{3}}{2}$ .
$\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}$
Add like terms
The denominators are equal, so add the numerators and retain the common denominator. The result is exact, so no rounding is needed.
$\frac{2\sqrt{3}}{2}=\sqrt{3}$

Answer:

\sqrt{3}

Check: Since each term is about

0.866

, their sum is about

1.732

, consistent with

\sqrt{3}

.

Common mistakes and how to avoid them

Giving cosine a positive sign in the second quadrant because its reference angle has positive cosine.

Correction: The reference angle gives the magnitude. The actual quadrant determines the sign; cosine is negative in the second quadrant.

Using a calculator decimal when an exact value is requested.

Correction: Write the exact fraction or surd first. Use a decimal only when the question asks for an approximation or as a check.

Entering radians while the calculator is set to degrees, or the reverse.

Correction: Check the angle mode before calculating. A mismatch can produce a plausible-looking but incorrect decimal.

Treating tangent as defined at

\frac{\pi}{2}

.

Correction: Tangent is sine divided by cosine, and cosine is zero at

\frac{\pi}{2}

. Division by zero is undefined.

Lesson summary

On the unit circle, the point at angle $\theta$ has coordinates $(\cos\theta,\sin\theta)$ .
Remember the standard exact values, then use reference angles and quadrant signs for other positions.
Use a calculator or graph to check signs and approximate size, while keeping exact reasoning visible.

Check your understanding

Question 1

What is the exact value of

\cos\left(\frac{2\pi}{3}\right)

?

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

Show answer and explanation

-\frac{1}{2}

The reference angle is

\frac{\pi}{3}

, whose cosine is

\frac{1}{2}

. The angle is in the second quadrant, where cosine is negative.

Question 2

What is the exact value of

\sin\left(\frac{3\pi}{2}\right)

?

$1$
$0$
$-1$
Undefined

Show answer and explanation

-1

At the bottom of the unit circle the point is

(0,-1)

, so its vertical coordinate, sine, is

-1

.

Question 3

A calculator is set to radians. Which sign should its value for

\cos\left(\frac{5\pi}{4}\right)

have?

Positive, because the angle is greater than $\pi$
Negative, because the angle is in the third quadrant
Zero, because the angle is halfway around the circle
Positive, because cosine is always positive

Show answer and explanation

Negative, because the angle is in the third quadrant

The angle is in the third quadrant, where the horizontal coordinate and hence cosine are negative.

Key terms

Unit circle: A circle of radius $1$ centred at the origin of a coordinate plane.
Reference angle: The acute angle between an angle's terminal side and the horizontal axis.
Exact value: A value written without rounding, often as an integer, fraction, or root.
Terminal side: The direction reached after turning through an angle from the positive horizontal direction.

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