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# TRIGONOMETRY: SINE AND COSINE RULE

CONTENT

1. Derivation and application of sine rule.
2. Derivation and application of cosine rule.

SINE RULE

Given any triangle ABC (acute or obtuse), with the angles labelled with capital letters A, B, C and the sides opposite these angles labelled with the corresponding small letters a, b, and c respectively as shown below.

The sine rule states that;

$$\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$$

OR

$$\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}$$

PROOF OF THE RULE

Using Acute – angled triangle

Given: Any ∆ABC with B acute.

To prove: $$\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$$

Construction: Draw the perpendicular from C to AB.

Proof: Using the lettering in the diagram above.

$$sinA = \frac{h}{b} \\ h = bsinA …….(i) \\ sinB = \frac{h}{a} \\ h = asinB …….(ii)$$

From equation (i) and (ii)

$$bsinA = asinB \\ ∴ \frac{a}{sinA} = \frac{b}{sinB}$$

Similarly, by drawing a perpendicular from B to AC

$$\frac{a}{sinA} = \frac{c}{sinC} \\ ∴ \frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$$

Q.E.D

Using Obtuse – angled triangle

Given: any ∆ABC with B obtuse

To Prove: $$\frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC}$$

Construction: Draw the perpendicular from C to AB produced.

Lesson tags: General Mathematics Lesson Notes, General Mathematics Objective Questions, SS2 General Mathematics, SS2 General Mathematics Evaluation Questions, SS2 General Mathematics Evaluation Questions Second Term, SS2 General Mathematics Objective Questions, SS2 General Mathematics Objective Questions Second Term, SS2 General Mathematics Second Term
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