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

Area of a triangle using sine?

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

When do you use the sine rule?

A side and its opposite angle are known, plus one more 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 an angle?

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

What is the ambiguous case?

Two sides and a non-included angle can give two different triangles.

Why can the sine rule give two angles?

Because \sin\theta=\sin(180^{\circ}-\theta).

How do you test the second solution in the ambiguous case?

Check that the angle sum with 180^{\circ}-\theta is still below 180^{\circ}.

When is there no triangle in the ambiguous case?

When the sine rule gives \sin\theta1.

Calculator rounding rule?

Keep full values and round only the final answer.

Sine and cosine rules and area of a triangleEdexcel International A Level Maths: Flashcards

Card 1 of 130 of 13 known

Question

State the sine rule.

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

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 $a$ and $b$
When do you use the sine rule?: A side and its opposite angle are known, plus one more 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 an angle?: The angle is obtuse (between $90^{\circ}$ and $180^{\circ}$ ).
What is the ambiguous case?: Two sides and a non-included angle can give two different triangles.
Why can the sine rule give two angles?: Because $\sin\theta=\sin(180^{\circ}-\theta)$ .
How do you test the second solution in the ambiguous case?: Check that the angle sum with $180^{\circ}-\theta$ is still below $180^{\circ}$ .
When is there no triangle in the ambiguous case?: When the sine rule gives $\sin\theta>1$ .
Which angle is opposite the longest side?: The largest angle.
Calculator rounding rule?: Keep full values and round only the final answer.

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

In triangle $PQR$ , $PQ=12$ cm, $PR=9$ cm and angle $QPR=40^{\circ}$ .
Find the size of angle $PQR$ .2 marks
A triangular plot of land $ABC$ has $AB=140$ m, $AC=95$ m and $BC=180$ m.
Find the size of the smallest angle of the plot.2 marks
In triangle $ABC$ , $AB=8.4$ cm, angle $ABC=52^{\circ}$ and angle $ACB=71^{\circ}$ .
Find the length of $AC$ .3 marks

See the full worksheet

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).