Basic Trigonometry (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Triangle $ABC$ has $AB=5$ cm, $BC=4$ cm and $AC=7$ cm.

Calculate the size of angle $BAC$, giving your answer to the nearest degree.

Triangle $PQR$ has $PQ=12$ cm, $QR=8$ cm and angle $PQR=60°$.

Calculate the length of $RP$, giving your answer correct to 3 significant figures.

The diagram shows triangle $ABC$ in which $AB=8.4$ cm, $BC=5.2$ cm and angle $ACB=28°$. Angle $BAC=x°$ and $AC=y$ cm.

Calculate the values of $x$ and $y$, giving your answers to 3 significant figures.

Two sides of a triangular garden are $2.8$ m and $12$ m long, and they meet at an angle of $67°$.

Calculate the area of the garden, giving your answer correct to 3 significant figures.

Calculate the length of the third side of the garden, giving your answer correct to 3 significant figures.

Triangle $ABC$ has sides of length $4$ cm, $5$ cm and $6$ cm.

(i) Show that the angle between the sides of length $4$ cm and $5$ cm has $\text{cos}\theta =\frac{1}{8}$.

(ii) Find the value of $\text{cos}\theta$ for each of the other two angles.

Two different triangles each have $AB=5.3$ cm, $BC=6.4$ cm and angle $ACB=38°$.

(i) Show that angle $BAC=132°$, correct to 3 significant figures, for one of the triangles.

(ii) Find angle $ABC$ for the other triangle.

A triangular field has an area of $10$ m². Two of its sides have lengths $3.2$ m and $8.4$ m, and the angle between them is $\theta$.

Show that one possible value of $\theta$ is $48.1°$, correct to 3 significant figures.

Show that there is another possible value of $\theta$. You must show all your working.

Using $\theta =48.1°$, calculate the perimeter of the field, giving your answer correct to 3 significant figures.

Triangle $ABC$ has $AB=x$ cm, $AC=(x+8)$ cm and angle $BAC=150°$. The area of the triangle is $2\frac{1}{4}$ cm².

Show that $x$ satisfies the equation $x^2+8x=9$.

Hence find the value of $x$.

Find the perimeter of the triangle, giving your answer correct to 3 significant figures.

Write the ratio of the angles of the triangle, to the nearest degree, in its simplest form.

The Save My Exams logo is a blue circle with a black lightning bolt through the centre. The lightning bolt is modelled as two congruent triangles $ABC$ and $DEF$ with $AB=DE=12$ cm, $BC=EF=15$ cm and $AC=DF=7$ cm.

Calculate the size of the angle $\alpha °$, giving your answer in degrees to 1 decimal place.

For the banner, the side lengths of the triangles are enlarged by a scale factor of $10$. Find the area the lightning bolt will take up on the banner, giving your answer in m².

Triangle $ABC$ has $AB=10$ cm and $AC=17$ cm, with angle $BAC$ acute. The area of the triangle is $40$ cm².

Find angle $BAC$, giving your answer to the nearest degree.

Show that the side length $BC$ is $9.43$ cm, correct to 3 significant figures.

Find angle $BCA$, giving your answer correct to 3 significant figures.

A point $D$ is placed on the side $AC$ so that $BD=BC$. Find the area of triangle $BCD$, giving your answer correct to 3 significant figures.

A triangle has side lengths $7.3$ cm, $9.8$ cm and $10.6$ cm.

Calculate the size of the largest angle.

A triangle has two sides of length $3$ m and $5$ m, with an angle of $45°$ between them.

Calculate the length of the third side, giving your answer correct to 3 significant figures.

A triangle has sides of length $4$ cm, $3$ cm and $2$ cm.

(i) Find the value of $\text{cos}\theta$ for each of the three angles.

(ii) State between which two sides the largest angle lies.

The diagram shows a quadrilateral field $ABCD$ in which alpacas are kept. $AB=128$ m, $BC=72$ m, $CD=120$ m, $DA=68$ m and angle $ABC=86°$.

Find the length $AC$, giving your answer to the nearest metre.

Find the angle between the fences $AD$ and $CD$.

Calculate the area of the field, giving your answer correct to 3 significant figures.

The diagram shows triangle $PQR$ with a point $S$ on $PR$. $PQ=13$ cm, $QR=5.8$ cm, $QS=y$ cm, angle $QSP=102°$, angle $QRS=37°$ and angle $QPS=x°$.

Calculate the values of $x$ and $y$, giving your answers to 3 significant figures.

Triangle $ABC$ has $AB=2x$ cm, $BC=(x+5)$ cm and $AC=7$ cm. The angle $ABC=60°$.

Find the value of $x$, giving your answer as a simplified surd.

The area of the triangle can be written in the form $a\sqrt{6}+b\sqrt{3}$, where $a$ and $b$ are integers. Find the values of $a$ and $b$.

Two different triangles each have $AB=2.8$ cm, $BC=3.5$ cm and angle $ACB=43°$.

Find the largest angle in each of the triangles.

A triangular field has an area of $157$ m². Two of its side lengths are $13.2$ m and $28.4$ m, and the angle $\theta$ lies between these two sides.

Find the two possible values of $\theta$.

Given that $\theta$ is acute, calculate the perimeter of the field, giving your answer to the nearest metre.

The diagram shows a quadrilateral field $ABCD$ in which unicorns are kept. $AB=145$ m, $BC=98$ m, $CD=215$ m, $DA=114$ m and angle $DAB=102°$.

Find the angle between the fences $BC$ and $CD$.

To be happy, each unicorn needs at least $1234$ m² of the field. Calculate the maximum number of unicorns that can be happily kept in the field.

Triangle $ABC$ has $AB=2x$ cm, $AC=(x+6)$ cm and angle $BAC=30°$. The area of the triangle is $5\frac{3}{4}$ cm².

Show that $2x^2+12x-23=0$.

Hence, or otherwise, find the perimeter of the triangle, giving your answer correct to 3 significant figures.

Given that angle $ABC$ is obtuse, find the ratio of the angles of the triangle, to the nearest degree.

A car company is designing a new metal logo. Points $A$, $B$, $C$ and $D$ all lie on the circumference of a circle. $AB$, $AC$, $AD$, $BC$ and $CD$ are chords, with $AB=CD$. The company wants the chords to satisfy $AB:BC:AC=2:4:5$.

The machine that makes the logo requires angle $ABC$ to be input, accurate to 1 decimal place. Calculate the angle $ABC$.

The acceptable error in angle $CAD$ is $1$ degree. Find the acceptable range for $x°$, giving your values to 1 decimal place.

Given that $BC=1$ inch, use the angles from parts (a) and (b) (correct to 1 decimal place) to calculate the area of the quadrilateral $ABCD$.

In triangle $ABC$, $AB=2x$ cm, $BC=10$ cm and $AC=(20-2x)$ cm. Angle $ABC=\theta$.

Show that $\text{cos}\theta =\frac{4x-15}{2x}$.

Given that $\text{cos}\theta =-\frac{1}{2}$, find the area of the triangle.

An artist is designing a triangular sculpture, made using three equal lengths of metal piping. When laid flat, the sculpture covers $21.8$ m².

Calculate the total length of metal piping needed. Give your answer to the nearest cm.

An isosceles triangle has two of its side lengths equal to $7.3$ cm and $9.8$ cm.

Calculate the difference between the two possible smallest angles.

Triangle $ABC$ has $AB=3x$ cm, $BC=5x$ cm and $AC=6x$ cm.

Calculate the size of angle $BAC$, giving your answer to 2 decimal places.

Given that the perimeter of the triangle is $37.8$ cm, find the area of the triangle, giving your answer correct to 3 significant figures.

Triangles $PSQ$ and $SQR$ are such that $PS=SQ=QR$, where $S$ lies on $PR$. $PQ=15$ cm, $QR=2x$ cm and angle $PSQ=120°$.

Calculate the exact value of $x$.

Calculate the area of triangle $PQR$, leaving your answer in surd form.

The diagram shows a field $ABCD$ in which unicorns are kept, where $AB$ is parallel to $CD$. $AB=318$ m, $CD=235$ m, $AD=142$ m and angle $DAB=92°$.

To be happy, each unicorn needs at least $2222$ m² of the field. Calculate the maximum number of unicorns that can happily be kept in the field.

Triangle $ABC$ has $AB=x$ cm, $BC=(4-x)$ cm, angle $BAC=\theta$ and angle $BCA=30°$.

Given that $\text{sin}\theta =\frac{1}{\sqrt{2}}$, show that $x=4(\sqrt{2}-1)$.

Triangle $ABC$ has $AB=3x$ cm, $AC=(x+5)$ cm and angle $BAC=150°$. The area of the triangle is $7\frac{1}{4}$ cm².

Find the ratio of the angles of the triangle, to the nearest degree, in its simplest form.