Algebraic side lengths - Sine and Cosine Rules

Edexcel P1 2021 June Q3

9 marks Standard +0.3

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

[diagram]

Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle $A B C$ with

$A B = p$ metres
$A C = q$ metres
$B C = 2 \sqrt { 2 }$ metres
angle $B A C = 60 ^ { \circ }$
1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side $A C$ is 2 metres longer than side $A B$, use algebra to find

the exact value of $p$,
the exact value of $q$. Using the answers to part (b),

calculate the exact area of the flower bed.

OCR C2 Q2

4 marks Standard +0.3

2. The diagram shows triangle $P Q R$ in which $P Q = x , P R = 7 - x , Q R = x + 1$ and $\angle P Q R = 60 ^ { \circ }$. Using the cosine rule, find the value of $x$.

Edexcel AS Paper 1 2022 June Q4

6 marks Standard +0.3

4. Figure 1 Figure 1 shows a sketch of triangle $A B C$ with $A B = ( x + 2 ) \mathrm { cm } , B C = ( 3 x + 10 ) \mathrm { cm }$, $A C = 7 x \mathrm {~cm}$, angle $B A C = 60 ^ { \circ }$ and angle $A C B = \theta ^ { \circ }$

1. Show that $17 x ^ { 2 } - 35 x - 48 = 0$
2. Hence find the value of $x$.
Hence find the value of $\theta$ giving your answer to one decimal place.

OCR PURE Q6

6 marks Standard +0.8

6 \includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-4_442_661_840_260} The diagram shows triangle $A B C$, with $A B = x \mathrm {~cm} , A C = y \mathrm {~cm}$ and angle $B A C = 60 ^ { \circ }$. It is given that the area of the triangle is $( x + y ) \sqrt { 3 } \mathrm {~cm} ^ { 2 }$.

Show that $4 x + 4 y = x y$. When the vertices of the triangle are placed on the circumference of a circle, $A C$ is a diameter of the circle.
Determine the value of $x$ and the value of $y$.

OCR MEI AS Paper 1 2021 November Q5

5 marks Standard +0.3

5 The diagram shows the triangle ABC in which $\mathrm { AC } = 13 \mathrm {~cm}$ and AB is the shortest side. The perimeter of the triangle is 32 cm . The area is $24 \mathrm {~cm} ^ { 2 }$ and $\sin \mathrm { B } = \frac { 4 } { 5 }$. Determine the lengths of AB and BC .

CAIE P1 2021 March Q10

8 marks Challenging +1.2

For the case where angle $B A C = \frac { 1 } { 6 } \pi$ radians, find $k$ correct to 4 significant figures.
For the general case in which angle $B A C = \theta$ radians, where $0 < \theta < \frac { 1 } { 2 } \pi$, it is given that $\frac { \theta } { \sin \theta } > 1$. Find the set of possible values of $k$.

AQA AS Paper 2 2022 June Q8

7 marks Standard +0.8

8 Triangle $A B C$ has sides of length $( m - n ) , m$ and $( m + n )$ where $0 < 2 n < m$ Angle $A$ is the largest angle in the triangle.
8

1. Explain why angle $A$ must be opposite the side of length $( m + n )$. 8
  1. (ii) Using the cosine rule, show that $\cos A = \frac { m - 4 n } { 2 ( m - n ) }$ 8
2. You are given that $B C$ is the diameter of a circle, and $A$ lies on the circumference of the circle. The value of $m$ is 8 Calculate the value of $n$.

Pre-U Pre-U 9794/1 2015 June Q3

3 marks Standard +0.3

3 \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_305_825_630_660} The diagram shows a triangle $A B C$ in which angle $B = 39 ^ { \circ }$, angle $C = 28 ^ { \circ } , A B = x \mathrm {~cm}$ and $A C = ( 2 x - 1 ) \mathrm { cm }$. Find the value of $x$.

Pre-U Pre-U 9794/1 Specimen Q4

4 marks Standard +0.8

4 The diagram shows triangle $A B C$, in which $A B = 1$ unit , $A C = k$ units and $B C = 2$ units .

Express $\cos C$ in terms of $k$.
Given that $\cos C < \frac { 7 } { 8 }$, show that $2 k ^ { 2 } - 7 k + 6 < 0$ and find the set of possible values of $k$.

Edexcel C2 Q2

4 marks Moderate -0.3

\includegraphics{figure_1} Figure 1 shows triangle $PQR$ in which $PQ = x$, $PR = 7 - x$, $QR = x + 1$ and $\angle PQR = 60°$. Using the cosine rule, find the value of $x$. [4]

OCR H240/03 2021 November Q2

6 marks Standard +0.3

\includegraphics{figure_2} The diagram shows triangle $ABC$ in which angle $A$ is $60°$ and the lengths of $AB$ and $AC$ are $(4 + h)$ cm and $(4 - h)$ cm respectively.

Show that the length of $BC$ is $p$ cm where $$p^2 = 16 + 3h^2.$$ [2]
Hence show that, when $h$ is small, $p \approx 4 + \lambda h^2 + \mu h^4$, where $\lambda$ and $\mu$ are rational numbers whose values are to be determined. [4]

WJEC Unit 1 Specimen Q13

7 marks Standard +0.3

In triangle $ABC$, $BC = 12$ cm and $\cos ABC = \frac{2}{3}$. The length of $AC$ is 2 cm greater than the length of $AB$.

Find the lengths of $AB$ and $AC$. [4]
Find the exact value of $\sin BAC$. Give your answer in its simplest form. [3]

SPS SPS FM 2020 October Q5

6 marks Moderate -0.3

\includegraphics{figure_5} The diagram shows triangle $ABC$, with $AB = x$ cm, $AC = (x + 2)$ cm, $BC = 2\sqrt{7}$ cm and angle $CAB = 60°$.

Find the value of $x$. [4]
Find the area of triangle $ABC$, giving your answer in an exact form as simply as possible. [2]