Right angled triangles appear in both two dimensional and three dimensional problems. Trigonometric relationships are used to calculate unknown sides or angles when enough information is given.

 

 

Right Angled Triangles in 2-D

In 2-D, a right angled triangle lies flat on a plane and contains one angle equal to
\( 90^\circ \).

 

To solve problems, you use the trigonometric ratios based on an angle \( \theta \):

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

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

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

 

The hypotenuse is always the longest side and is opposite the right angle.

 

 

Calculating a Side in 2-D

If an angle and one side are known, a missing side can be found.

 

Example
A right angled triangle has an angle of
\( 40^\circ \)
and an adjacent side of length 7 cm.

 

To find the hypotenuse:
$$
\cos 40^\circ = \frac{7}{h}
$$

 

Rearranging:
$$
h = \frac{7}{\cos 40^\circ}
$$

 

 

Calculating an Angle in 2-D

If two sides are known, an angle can be calculated using inverse trigonometric functions.

 

Example
The opposite side is 5 cm and the adjacent side is 12 cm.

 

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

$$
\theta = \tan^{-1}\left(\frac{5}{12}\right)
$$

 

Angles should be given to an appropriate degree of accuracy.

 

 

Right Angled Triangles in 3-D

In 3-D problems, the triangle may be inside a solid rather than lying flat.

 

Common examples include:
• diagonals of cuboids
• sloping edges of pyramids
• distances between opposite corners

 

The triangle used must still be right angled, even if it is not immediately obvious.

 

Always identify the right angled triangle before using trigonometry

 

 

Calculating a Side in 3-D

3-D problems often involve two steps.

 

First, find a length on a face of the shape.
Then use that length in a second right angled triangle.

 

Example
A cuboid has base dimensions 6 cm by 8 cm and a height of 10 cm.

 

Step 1: Find the diagonal of the base
$$
6^2 + 8^2 = d^2
$$

$$
d = 10
$$

 

Step 2: Use trigonometry or Pythagoras with the height
$$
10^2 + 10^2 = x^2
$$

$$
x = \sqrt{200}
$$

 

 

Calculating an Angle in 3-D

Angles in 3-D are often found between:
• a diagonal and the base
• a sloping edge and the horizontal

 

Once the right angled triangle is identified, trigonometry is used in the same way as in 2-D.

 

Example
If the vertical height is 5 m and the horizontal distance is 12 m:

 

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

$$
\theta = \tan^{-1}\left(\frac{5}{12}\right)
$$

 

This angle could represent an angle of elevation or angle of inclination.

 

 

Common Errors to Avoid

Common mistakes include:
• using trigonometry without a right angle
• choosing the wrong triangle in 3-D
• mixing up opposite and adjacent sides
• forgetting to use inverse functions for angles

 

Clear diagrams help prevent these errors

 

 

Key Points to Remember

Trigonometry only applies to right angled triangles.
SOH CAH TOA helps choose the correct ratio.
Inverse functions are used to find angles.
3-D problems require identifying the correct internal triangle.
Clear diagrams and step by step working improve accuracy.

 

Being able to calculate sides and angles in both 2-D and 3-D right angled triangles allows complex geometric problems to be solved confidently and accurately.