Trigonometry can be used to solve problems involving any angle, not just right angles. This includes situations in two dimensions and three dimensions, using the sine rule and cosine rule where appropriate.

 

 

When to Use the Sine Rule or Cosine Rule

The sine rule and cosine rule are used for non right angled triangles.

 

Use the sine rule when:
• two angles and one side are known
• two sides and a non included angle are known

 

Use the cosine rule when:
• two sides and the included angle are known
• all three sides are known and an angle is required

 

Always sketch the triangle and label known values first

 

 

The Sine Rule

The sine rule relates the sides of a triangle to the sines of the opposite angles.

$$
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
$$

 

Each side is opposite its corresponding angle.

 

 

Using the Sine Rule to Find a Side

If two angles and one side are known, the sine rule can be rearranged to find a missing side.

 

Example structure
$$
\frac{a}{\sin A} = \frac{b}{\sin B}
$$

 

Rearrange to make the required side the subject.

 

 

Using the Sine Rule to Find an Angle

If two sides and one non included angle are known, the sine rule can be used to find an angle.

 

In some cases, there may be two possible angles that satisfy the equation.

 

This is known as the ambiguous case.

 

Always check whether a second solution is possible

 

 

The Cosine Rule

The cosine rule is an extension of Pythagoras’ theorem.

 

It is used when the triangle is not right angled.

$$
c^2 = a^2 + b^2 - 2ab\cos C
$$

 

The formula can be rearranged depending on which side or angle is required.

 

 

Using the Cosine Rule to Find a Side

If two sides and the included angle are known, substitute directly into the formula.

 

Example structure
$$
c^2 = a^2 + b^2 - 2ab\cos C
$$

 

After calculating \( c^2 \), take the square root to find \( c \).

 

 

Using the Cosine Rule to Find an Angle

If all three sides are known, the cosine rule can be rearranged to find an angle.

 

Example structure
$$
\cos C = \frac{a^2 + b^2 - c^2}{2ab}
$$

 

Then use the inverse cosine function to find the angle.

 

 

Applying Trigonometry in 2-D Problems

In 2-D problems, triangles may include:
• obtuse angles
• reflex angles
• multiple triangles joined together

 

You may need to:
• find missing sides or angles
• split shapes into triangles
• apply the sine or cosine rule more than once

 

Angles greater than \( 90^\circ \) are handled in the same way.

 

 

Applying Trigonometry in 3-D Problems

In 3-D problems, triangles are often inside solids.

 

To solve these problems:
• identify a suitable triangle
• find face diagonals or base lengths first
• apply the sine or cosine rule as needed

 

Some problems require a combination of:
• Pythagoras’ theorem
• sine rule
• cosine rule

 

Work step by step and label all lengths clearly

 

 

Common Errors to Avoid

Common mistakes include:
• using SOH CAH TOA in non right angled triangles
• choosing the wrong rule
• mixing up opposite sides and angles
• forgetting that angles can be obtuse

 

A clear diagram reduces most errors.

 

 

Key Points to Remember

The sine rule and cosine rule apply to any triangle.
Use the sine rule when an angle opposite a side is known.
Use the cosine rule with two sides and the included angle or three sides.
Some sine rule problems have two possible solutions.
Trigonometry can be applied in both 2-D and 3-D contexts.

 

Applying trigonometry with angles of any size allows complex geometric problems to be solved accurately using consistent mathematical methods.