Sec 3 E-Math · Further trigonometry

Cosine rule questions with answers

Practice questions on cosine rule, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

In triangle $ABC$ , $AB = 12$ cm, $AC = 7$ cm and angle $BAC = 42^\circ$ . Find the length of $BC$ , correct to 3 significant figures.

Step 1 of 4

Which rule links the sides and angles you know with the one you want?

\text{Cosine rule}

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Questions

Question 1Find a side: acute angle
In triangle $ABC$ , $AB = 14$ cm, $AC = 8$ cm and angle $BAC = 63^\circ$ . Find the length of $BC$ , correct to 3 significant figures.
Show the worked solution
1. Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
2. Step 2: Write the rule, then name b, c and A $a^2 = b^2 + c^2 - 2bc \cos A$ $b = 8$ , $c = 14$ , $A = 63^\circ$
3. Step 3: Substitute $x^2 = 8^2 + 14^2 - 2(8)(14) \cos 63^\circ$ $x^2 = 158.3061\ldots$
4. Step 4: Square root $x = 12.5819\ldots$ $\phantom{x} = 12.6$ cm (3 s.f.)
Question 2Find a side: obtuse angle
In triangle $ABC$ , $AB = 10$ cm, $AC = 6$ cm and angle $BAC = 125^\circ$ . Find the length of $BC$ , correct to 3 significant figures.
Show the worked solution
1. Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
2. Step 2: Write the rule, then name b, c and A $a^2 = b^2 + c^2 - 2bc \cos A$ $b = 6$ , $c = 10$ , $A = 125^\circ$
3. Step 3: Substitute $x^2 = 6^2 + 10^2 - 2(6)(10) \cos 125^\circ$ $x^2 = 204.8291\ldots$
4. Step 4: Square root $x = 14.3118\ldots$ $\phantom{x} = 14.3$ cm (3 s.f.)
Question 3Find an angle from three sides
In triangle $ABC$ , $BC = 8$ cm, $AC = 5$ cm and $AB = 12$ cm. Find angle $BAC$ , correct to 1 decimal place.
Show the worked solution
1. Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
2. Step 2: Write the rule with cos A as the subject, then name a, b and c $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ $a = 8$ (the side opposite the angle), $b = 5$ , $c = 12$
3. Step 3: Substitute $\cos \theta = \dfrac{5^2 + 12^2 - 8^2}{2(5)(12)}$ $\phantom{\cos \theta} = 0.875$
4. Step 4: Use cos⁻¹ $\theta = \cos^{-1}(0.875)$ $\theta = 28.9550\ldots^\circ$ $\phantom{\theta} = 29.0^\circ$ (1 d.p.)
Question 4Find the obtuse angle
In triangle $ABC$ , $BC = 14$ cm, $AC = 8$ cm and $AB = 9$ cm. Find angle $BAC$ , correct to 1 decimal place.
Show the worked solution
1. Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
2. Step 2: Write the rule with cos A as the subject, then name a, b and c $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ $a = 14$ (the side opposite the angle), $b = 8$ , $c = 9$
3. Step 3: Substitute $\cos \theta = \dfrac{8^2 + 9^2 - 14^2}{2(8)(9)}$ $\phantom{\cos \theta} = -0.3541\ldots$
4. Step 4: Use cos⁻¹: a negative cosine gives an obtuse angle $\theta = \cos^{-1}(-0.3541\ldots)$ $\theta = 110.7423\ldots^\circ$ $\phantom{\theta} = 110.7^\circ$ (1 d.p.)

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Worked example

In triangle

ABC

AB = 12

cm,

AC = 7

cm and angle

BAC = 42^\circ

. Find the length of

BC

, correct to 3 significant figures.

Step 1 of 4

Which rule links the sides and angles you know with the one you want?

\text{Cosine rule}

Questions

Question 1Find a side: acute angle

In triangle $ABC$ , $AB = 14$ cm, $AC = 8$ cm and angle $BAC = 63^\circ$ . Find the length of $BC$ , correct to 3 significant figures.

Show the worked solution

Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
Step 2: Write the rule, then name b, c and A $a^2 = b^2 + c^2 - 2bc \cos A$ $b = 8$ , $c = 14$ , $A = 63^\circ$
Step 3: Substitute $x^2 = 8^2 + 14^2 - 2(8)(14) \cos 63^\circ$ $x^2 = 158.3061\ldots$
Step 4: Square root $x = 12.5819\ldots$ $\phantom{x} = 12.6$ cm (3 s.f.)

Question 2Find a side: obtuse angle

In triangle $ABC$ , $AB = 10$ cm, $AC = 6$ cm and angle $BAC = 125^\circ$ . Find the length of $BC$ , correct to 3 significant figures.

Show the worked solution

Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
Step 2: Write the rule, then name b, c and A $a^2 = b^2 + c^2 - 2bc \cos A$ $b = 6$ , $c = 10$ , $A = 125^\circ$
Step 3: Substitute $x^2 = 6^2 + 10^2 - 2(6)(10) \cos 125^\circ$ $x^2 = 204.8291\ldots$
Step 4: Square root $x = 14.3118\ldots$ $\phantom{x} = 14.3$ cm (3 s.f.)

Question 3Find an angle from three sides

In triangle $ABC$ , $BC = 8$ cm, $AC = 5$ cm and $AB = 12$ cm. Find angle $BAC$ , correct to 1 decimal place.

Show the worked solution

Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
Step 2: Write the rule with cos A as the subject, then name a, b and c $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ $a = 8$ (the side opposite the angle), $b = 5$ , $c = 12$
Step 3: Substitute $\cos \theta = \dfrac{5^2 + 12^2 - 8^2}{2(5)(12)}$ $\phantom{\cos \theta} = 0.875$
Step 4: Use cos⁻¹ $\theta = \cos^{-1}(0.875)$ $\theta = 28.9550\ldots^\circ$ $\phantom{\theta} = 29.0^\circ$ (1 d.p.)

Question 4Find the obtuse angle

In triangle $ABC$ , $BC = 14$ cm, $AC = 8$ cm and $AB = 9$ cm. Find angle $BAC$ , correct to 1 decimal place.

Show the worked solution

Step 1: Which rule links the sides and angles you know with the one you want? $\text{Cosine rule}$
Step 2: Write the rule with cos A as the subject, then name a, b and c $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ $a = 14$ (the side opposite the angle), $b = 8$ , $c = 9$
Step 3: Substitute $\cos \theta = \dfrac{8^2 + 9^2 - 14^2}{2(8)(9)}$ $\phantom{\cos \theta} = -0.3541\ldots$
Step 4: Use cos⁻¹: a negative cosine gives an obtuse angle $\theta = \cos^{-1}(-0.3541\ldots)$ $\theta = 110.7423\ldots^\circ$ $\phantom{\theta} = 110.7^\circ$ (1 d.p.)