Three Important Angles Revision Notes for Leaving Cert Mathematics

Three Important Angles

Introduction

In trigonometry, the angles 30°, 45°, and 60° are considered special because they appear frequently in calculations and allow us to find exact trigonometric values without using a calculator. These angles produce exact values that can be expressed as simple fractions or surds (expressions involving square roots).

Understanding these three important angles is essential for Leaving Cert success, as they often appear in exam questions where calculator use may be restricted or where exact answers are required.

The 45° angle

The 45° angle comes from a special type of triangle called an isosceles right triangle. This triangle has two equal sides of length 1 unit each, making it perfectly symmetrical.

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In an isosceles right triangle:

The two equal sides each measure 1 unit
The hypotenuse measures $\sqrt{2}$ units (found using Pythagoras' theorem: $\sqrt{1^2 + 1^2} = \sqrt{2}$ )
Both base angles are 45°

From this triangle construction, we can determine the exact trigonometric ratios for 45°:

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

Notice that sine and cosine have the same value for 45°, which makes sense because the opposite and adjacent sides are equal in this isosceles triangle.

The 30° and 60° angles

The 30° and 60° angles are found together in what's called a 30-60-90 triangle. This is a right-angled triangle where the sides are in the ratio 1 : √3 : 2.

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In a 30-60-90 triangle:

The shortest side (opposite to the 30° angle) has length 1
The middle side (opposite to the 60° angle) has length $\sqrt{3}$
The hypotenuse has length 2

From this triangle, we can find the exact ratios for both 30° and 60°:

For 60°:

$\sin 60° = \frac{\sqrt{3}}{2}$
$\cos 60° = \frac{1}{2}$
$\tan 60° = \sqrt{3}$

For 30°:

$\sin 30° = \frac{1}{2}$
$\cos 30° = \frac{\sqrt{3}}{2}$
$\tan 30° = \frac{1}{\sqrt{3}}$

Summary table of exact values

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

Worked example 1

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Worked Example: Finding sides in a 30-60-90 triangle

Question: Without using a calculator, find the values of x and y in the right-angled triangle where one angle is 60°, and the adjacent side to this angle is 4 units.

Solution:

To find x (opposite side):

We use $\tan 60° = \frac{\text{opposite}}{\text{adjacent}}$
$\frac{x}{4} = \tan 60°$
$\frac{x}{4} = \sqrt{3}$
Therefore: $x = 4\sqrt{3}$

To find y (hypotenuse):

We use $\cos 60° = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\frac{4}{y} = \cos 60°$
$\frac{4}{y} = \frac{1}{2}$
Therefore: $y = 8$

Worked example 2

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Worked Example: Calculating exact values

Question: Find the exact value of $\sin^2 30° + \cos^2 45°$

Solution:

First, find each value:

$\sin 30° = \frac{1}{2}$ , so $\sin^2 30° = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$
$\cos 45° = \frac{1}{\sqrt{2}}$ , so $\cos^2 45° = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}$

Therefore: $\sin^2 30° + \cos^2 45° = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}$

Worked example 3

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Worked Example: Proving trigonometric identities

Question: Without a calculator, show that $\tan 30° \times \tan 60° = 1$

Solution:

We know that:

$\tan 30° = \frac{1}{\sqrt{3}}$
$\tan 60° = \sqrt{3}$

Therefore: $\tan 30° \times \tan 60° = \frac{1}{\sqrt{3}} \times \sqrt{3} = \frac{\sqrt{3}}{\sqrt{3}} = 1$ ✓

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Exam Tips

These exact values are provided in the Formulae and Tables booklet (page 13)
Always express answers in their simplest surd form when exact values are required
Remember that 30° and 60° are complementary angles (add to 90°), so their sine and cosine values are "swapped"
For 45°, sine and cosine are always equal
Practice recognising when these special angles appear in more complex problems

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Common Mistakes to Avoid

Don't confuse the exact values - memorise the key ratios or know how to derive them quickly
Remember that $\frac{1}{\sqrt{3}}$ can be rationalised to $\frac{\sqrt{3}}{3}$ if required
Be careful with the signs - these angles are all in the first quadrant so all ratios are positive

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Key Points to Remember:

The three important angles (30°, 45°, 60°) give exact trigonometric values that don't require a calculator
45° comes from an isosceles right triangle with sides 1:1:√2
30° and 60° come from a 30-60-90 triangle with sides 1:√3:2
For 45°: $\sin = \cos = \frac{1}{\sqrt{2}}$ , and $\tan = 1$
For 30° and 60°: their sine and cosine values are complementary (swapped)

Three Important Angles (Leaving Cert Mathematics): Revision Notes