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

Area of a triangle using sine?

\frac12ab\sin C, with C between sides a and b.

When do you use the sine rule?

When you know a side and its opposite angle, plus one other side or angle.

When do you use the cosine rule?

Two sides and the included angle, or all three sides.

What does a negative cosine tell you about a triangle angle?

The angle is obtuse (between 90^\circ and 180^\circ).

Why can the sine rule give two possible angles?

Because \sin\theta=\sin(180^\circ-\theta); use the angle sum and longest-side rule to decide.

Which angle is the smallest in a triangle?

The one opposite the shortest side.

Sine and cosine rules and area of a triangleAQA A-Level Maths: Flashcards

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State the sine rule.

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State the sine rule.: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
State the cosine rule for a side.: $a^2=b^2+c^2-2bc\cos A$
State the cosine rule for an angle.: $\cos A=\frac{b^2+c^2-a^2}{2bc}$
Area of a triangle using sine?: $\frac12ab\sin C$ , with $C$ between sides $a$ and $b$ .
When do you use the sine rule?: When you know a side and its opposite angle, plus one other side or angle.
When do you use the cosine rule?: Two sides and the included angle, or all three sides.
$\sin(180^\circ-\theta)$ and $\cos(180^\circ-\theta)$ ?: $\sin\theta$ and $-\cos\theta$
What does a negative cosine tell you about a triangle angle?: The angle is obtuse (between $90^\circ$ and $180^\circ$ ).
Exact values of $\sin150^\circ$ and $\cos120^\circ$ ?: $\frac12$ and $-\frac12$
Why can the sine rule give two possible angles?: Because $\sin\theta=\sin(180^\circ-\theta)$ ; use the angle sum and longest-side rule to decide.
Which angle is the smallest in a triangle?: The one opposite the shortest side.
How are the bearings of $A$ from $B$ and $B$ from $A$ related?: They differ by $180^\circ$ .

Exam questions on Sine and cosine rules and area of a triangle

In triangle $ABC$ , $AB=9$ cm, $BC=14$ cm and $A\hat{B}C=40^\circ$ .
Find the size of angle $B\hat{A}C$ , to 1 decimal place.2 marks
In triangle $LMN$ , $LM=6$ m, $LN=10$ m and $MN=14$ m.
Find the size of the smallest angle of the triangle, to 1 decimal place.2 marks
Points $A$ and $B$ are $120$ m apart on a straight road on level ground. A tower $T$ stands in a field, with $T\hat{A}B=52^\circ$ and $T\hat{B}A=71^\circ$ . A calculator may be used.
Find the distance $AT$ , to 3 significant figures.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).