Exact Trigonometric Values Revision Notes for HSC SSCE Mathematics Advanced

Exact Trigonometric Values

Introduction to trigonometric ratios

In a right-angled triangle, we can describe the relationship between the sides using special ratios called trigonometric ratios. The three main ratios are sine, cosine, and tangent, which relate to an angle $\theta$ in the triangle.

The three trigonometric ratios are defined as:

$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

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SOHCAHTOA Mnemonic

A helpful way to remember these ratios is the mnemonic SOHCAHTOA:

SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

This mnemonic is your best friend when working with trigonometric ratios!

Pythagoras' theorem

Before we derive exact values, it's important to recall Pythagoras' theorem. For a right-angled triangle with hypotenuse $c$ and other sides $a$ and $b$ , the theorem states:

$c^2 = a^2 + b^2$

This fundamental relationship helps us calculate unknown side lengths when we know two sides of a right-angled triangle.

Deriving exact values for 45° angles

To find the exact trigonometric values for $45°$ , we use a special triangle called an isosceles right-angled triangle. This triangle has two equal sides of length 1 unit and two equal base angles of $45°$ each.

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Worked Example: Deriving the 45° Exact Values

Using Pythagoras' theorem, we can find the hypotenuse:

$\text{Hypotenuse}^2 = 1^2 + 1^2 = 2$

$\text{Hypotenuse} = \sqrt{2}$

Now we can calculate the exact trigonometric ratios for $45°$ :

$\sin 45° = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}$

$\cos 45° = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}$

$\tan 45° = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{1} = 1$

Rationalising the denominator

When working with exact values, we prefer to write answers in rationalised form, which means removing any square roots from the denominator.

To rationalise $\frac{1}{\sqrt{2}}$ , we multiply both the numerator and denominator by $\sqrt{2}$ :

$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Therefore, the exact values for 45° in rationalised form are:

$\sin 45° = \frac{\sqrt{2}}{2}$

$\cos 45° = \frac{\sqrt{2}}{2}$

$\tan 45° = 1$

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Universal Property of Similar Triangles

The side lengths in this triangle are $1$ , $1$ , and $\sqrt{2}$ , but these exact ratios apply to any isosceles right-angled triangle, regardless of size. The angles will always be $45°$ , $45°$ , and $90°$ , so the ratios remain the same.

Deriving exact values for 30° and 60° angles

To find the exact values for $30°$ and $60°$ angles, we start with an equilateral triangle where all sides measure $2$ units and all angles measure $60°$ .

Next, we draw a line from the top vertex to the base, creating a perpendicular bisector. This divides the equilateral triangle into two congruent right-angled triangles.

Each right-angled triangle now has:

A base of $1$ unit (half of the original base)
A hypotenuse of $2$ units (the original side)
Two angles of $30°$ and $60°$

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Worked Example: Finding the Height of the Triangle

Using Pythagoras' theorem, we can find the height:

$2^2 = 1^2 + h^2$

$4 = 1 + h^2$

$h^2 = 3$

$h = \sqrt{3}$

Now we have a 30°-60°-90° triangle with sides $1$ , $\sqrt{3}$ , and $2$ . We can determine the exact trigonometric values:

For the 30° angle:

$\sin 30° = \frac{1}{2}$

$\cos 30° = \frac{\sqrt{3}}{2}$

$\tan 30° = \frac{1}{\sqrt{3}}$

For the 60° angle:

$\sin 60° = \frac{\sqrt{3}}{2}$

$\cos 60° = \frac{1}{2}$

$\tan 60° = \sqrt{3}$

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Universal Property of 30-60-90 Triangles

Just like with the $45°$ triangle, any right-angled triangle with angles of 30° and 60° will have the same trigonometric ratios, regardless of the actual side lengths. The ratios are determined by the angles, not the size!

Complementary angles

Two angles are called complementary angles if they add up to $90°$ . There is a special relationship between the sine and cosine of complementary angles.

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The Complementary Angle Relationship

For any angle $\theta$ :

$\sin \theta = \cos(90° - \theta)$

$\cos \theta = \sin(90° - \theta)$

This means that the sine of an angle equals the cosine of its complement, and vice versa.

Let's see this relationship in action with our exact values:

$\sin 30° = \cos 60°$ (both equal $\frac{1}{2}$ )
$\sin 60° = \cos 30°$ (both equal $\frac{\sqrt{3}}{2}$ )
$\sin 45° = \cos 45°$ (both equal $\frac{\sqrt{2}}{2}$ )

Reference triangles

Here are the two special triangles side by side for easy reference:

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Special Triangle Ratios

The 45-45-90 triangle has sides in the ratio $1 : 1 : \sqrt{2}$
The 30-60-90 triangle has sides in the ratio $1 : \sqrt{3} : 2$

These ratios are crucial for deriving exact values quickly!

Summary of exact values

Here is a complete table of exact trigonometric values for common angles:

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Key Exact Values Table

Angle	sin	cos	tan
$30°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45°$	$\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$	$\frac{1}{\sqrt{2}}$ or $\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

Special Cases at Boundary Angles:

Angle	sin	cos	tan
$0°$	$0$	$1$	$0$
$90°$	$1$	$0$	Undefined

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Exam Tip: Rationalisation

When presenting answers in surd form, always give the final result in rationalised form. This means no square roots should appear in the denominator. Examiners expect to see $\frac{\sqrt{2}}{2}$ rather than $\frac{1}{\sqrt{2}}$ .

Worked example 1: Finding cos 45°

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Worked Example: Finding cos 45°

Question: What is the exact value of $\cos 45°$ ?

Solution:

We use the $45°$ - $45°$ - $90°$ triangle to find this value.

Write the cosine ratio:

$\cos 45° = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

Substitute the values from the triangle:

$\cos 45° = \frac{1}{\sqrt{2}}$

Rationalise the denominator by multiplying by $\frac{\sqrt{2}}{\sqrt{2}}$ :

$\cos 45° = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$

Perform the multiplication and simplify:

$\cos 45° = \frac{\sqrt{2}}{2}$

Therefore, the exact value is $\cos 45° = \frac{\sqrt{2}}{2}$ .

Worked example 2: Finding cos 60°

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Worked Example: Finding cos 60°

Question: What is the exact value of $\cos 60°$ ?

Solution:

We use the $30°$ - $60°$ - $90°$ triangle to find this value.

Write the cosine ratio:

$\cos 60° = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

Substitute the values from the triangle:

$\cos 60° = \frac{1}{2}$

The answer is already in simplest form, so the exact value is $\cos 60° = \frac{1}{2}$ .

Worked example 3: Finding an angle from its tangent

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Worked Example: Finding an Angle from its Tangent

Question: Find the angle $\theta$ in a right-angled triangle where $\tan \theta = \frac{1}{\sqrt{3}}$ .

Solution:

We need to identify which special angle has a tangent value of $\frac{1}{\sqrt{3}}$ .

From the $30°$ - $60°$ - $90°$ triangle, we know that:

$\tan 30° = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{\sqrt{3}}$

This matches our given value, so the answer is $\theta = 30°$ .

Remember!

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Key Points to Remember:

The three main trigonometric ratios are remembered using SOHCAHTOA: $\sin = \frac{\text{O}}{\text{H}}$ , $\cos = \frac{\text{A}}{\text{H}}$ , $\tan = \frac{\text{O}}{\text{A}}$
Exact values for 45° angles come from an isosceles right triangle with sides $1$ , $1$ , $\sqrt{2}$
Exact values for 30° and 60° angles come from splitting an equilateral triangle, creating a right triangle with sides $1$ , $\sqrt{3}$ , $2$
Complementary angles satisfy: $\sin \theta = \cos(90° - \theta)$ and $\cos \theta = \sin(90° - \theta)$
Always rationalise denominators in your final answers by removing square roots from the bottom of fractions
Memorise the exact values table for quick reference in exams, or know how to derive them from the special triangles

Exact Trigonometric Values (HSC SSCE Mathematics Advanced): Revision Notes