MYP Extended Mathematics Topic Sine Rule Revision Notes

MYP Extended Mathematics Topic Sine Rule Revision Notes | RevisionDojo

Using A Consistent Triangle Labeling Makes Formulas Work

When working with non-right-angled triangles, a standard labeling convention helps everyone communicate clearly and use the same formulas correctly.
Label the vertices with capital letters $A$, $B$, $C$.
The interior angle at each vertex uses the same letter ($\angle A$, $\angle B$, $\angle C$).
The side opposite each angle uses the same letter in lowercase ($a$, $b$, $c$).

Note

This convention is not strictly required, but it prevents common mistakes, especially when using the sine rule and cosine rule where "opposite" matters.

The Sine Rule Links Sides To Their Opposite Angles

The sine rule (also called the sine law) is a relationship that is true in every triangle (not just right-angled ones).
It says that the ratio "side divided by sine of its opposite angle" is constant within a triangle.

Definition

Sine rule

For any triangle $ABC$, $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

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

Hint

You almost never use all three fractions at once.
In problem solving, you select two matching pairs (a side and its opposite angle) and set their ratios equal, for example: $$\frac{a}{\sin A}=\frac{b}{\sin B} \quad \text{or} \quad \frac{\sin B}{b}=\frac{\sin C}{c}$$

Why The Sine Rule Is True (Idea Using A Perpendicular Height)

A useful way to understand the sine rule is to drop a perpendicular height in the triangle and connect it to right-triangle trigonometry.

Let $h$ be the height from $A$ to $BC$.
1. In right triangle $ABD$, $\sin B = \dfrac{h}{c}$, so $h=c\sin B$.
2. In right triangle $ACD$, $\sin C = \dfrac{h}{b}$, so $h=b\sin C$.
Since both expressions equal the same height $h$: $$c\sin B=b\sin C$$
Divide both sides by $bc$: $$\frac{\sin B}{b}=\frac{\sin C}{c}$$
Repeating this idea with a different altitude gives the full statement $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

Hint

If you remember only one idea, remember this: the sine rule comes from "the same height can be written two ways," using $$\sin(\text{angle})=\frac{\text{opposite}}{\text{hypotenuse}}$$ in two right triangles.

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Using A Consistent Triangle Labeling Makes Formulas Work

When working with non-right-angled triangles, a standard labeling convention helps everyone communicate clearly and use the same formulas correctly.
Label the vertices with capital letters $A$, $B$, $C$.
The interior angle at each vertex uses the same letter ($\angle A$, $\angle B$, $\angle C$).
The side opposite each angle uses the same letter in lowercase ($a$, $b$, $c$).

Note

This convention is not strictly required, but it prevents common mistakes, especially when using the sine rule and cosine rule where "opposite" matters.

The Sine Rule Links Sides To Their Opposite Angles

The sine rule (also called the sine law) is a relationship that is true in every triangle (not just right-angled ones).
It says that the ratio "side divided by sine of its opposite angle" is constant within a triangle.

Definition

Sine rule

For any triangle $ABC$, $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

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

Hint

You almost never use all three fractions at once.
In problem solving, you select two matching pairs (a side and its opposite angle) and set their ratios equal, for example: $$\frac{a}{\sin A}=\frac{b}{\sin B} \quad \text{or} \quad \frac{\sin B}{b}=\frac{\sin C}{c}$$

Why The Sine Rule Is True (Idea Using A Perpendicular Height)

A useful way to understand the sine rule is to drop a perpendicular height in the triangle and connect it to right-triangle trigonometry.

Let $h$ be the height from $A$ to $BC$.
1. In right triangle $ABD$, $\sin B = \dfrac{h}{c}$, so $h=c\sin B$.
2. In right triangle $ACD$, $\sin C = \dfrac{h}{b}$, so $h=b\sin C$.
Since both expressions equal the same height $h$: $$c\sin B=b\sin C$$
Divide both sides by $bc$: $$\frac{\sin B}{b}=\frac{\sin C}{c}$$
Repeating this idea with a different altitude gives the full statement $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

Hint

Definition

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Note

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Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Sine Rule Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Using A Consistent Triangle Labeling Makes Formulas Work

The Sine Rule Links Sides To Their Opposite Angles

Why The Sine Rule Is True (Idea Using A Perpendicular Height)

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Using A Consistent Triangle Labeling Makes Formulas Work

The Sine Rule Links Sides To Their Opposite Angles

Why The Sine Rule Is True (Idea Using A Perpendicular Height)

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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