GCSE / IGCSE Maths – Trigonometry

Write down the exact value of:(a) $\sin 30^\circ$(b) $\cos 45^\circ$(c) $\tan 60^\circ$(d) $\cos 90^\circ$

A right-angled triangle has a hypotenuse of $10$ cm and one angle of $30^\circ$. Find the exact lengths of the other two sides.

In triangle $PQR$, angle $Q = 90^\circ$, $PQ = 3$ cm and $QR = 4$ cm.(a) Find $PR$.(b) Write down the value of $\tan P$.(c) Write down the value of $\sin R$.

The graph of $y = \sin x$ is drawn for $0^\circ \le x \le 360^\circ$.(a) Write down the maximum value of $y$ and the value of $x$ at which it occurs.(b) Write down the two values of $x$ in this range for which $\sin x = 0$, other than $x = 0^\circ$.(c) Given that $\sin 50^\circ \approx 0.766$, use the symmetry of the graph to write down another solution of $\sin x = 0.766$ in the range $0^\circ \le x \le 360^\circ$.

(a) A right-angled triangle has hypotenuse $8$ cm and one angle of $35^\circ$. Find the length of the side opposite the $35^\circ$ angle.(b) In a different right-angled triangle, the side adjacent to an angle of $52^\circ$ is $5$ cm. Find the hypotenuse.(c) A ramp rises $7$ m over a horizontal distance of $12$ m. Find the angle the ramp makes with the horizontal.

A ladder of length $6$ m leans against a vertical wall, making an angle of $70^\circ$ with the horizontal ground. Find how far the foot of the ladder is from the wall, giving your answer to 2 decimal places.

In triangle $ABC$, angle $A = 38^\circ$, angle $B = 71^\circ$ and $BC = 9$ cm.(a) Find the length of $AC$.(b) Find the size of angle $C$.

In triangle $XYZ$, $XY = 7$ cm, $XZ = 10$ cm and angle $X = 58^\circ$.(a) Find the length of $YZ$.(b) Find the area of triangle $XYZ$.

A triangle has sides of length $8$ cm, $11$ cm and $14$ cm. Find the size of the largest angle.

A cuboid has length $8$ cm, width $6$ cm and height $5$ cm. Vertex $A$ is at one bottom corner and vertex $G$ is the top corner diagonally opposite.(a) Find the length of the base diagonal $AC$.(b) Find the length of the space diagonal $AG$.(c) Find the angle that $AG$ makes with the base.

A ship sails $12$ km from port $P$ on a bearing of $040^\circ$ to point $Q$. It then sails $9$ km on a bearing of $150^\circ$ to point $R$.(a) Show that angle $PQR = 70^\circ$.(b) Find the distance $PR$.(c) Find the bearing of $R$ from $P$.

A square-based pyramid has base side $10$ cm and vertical height $9$ cm. The apex $V$ is directly above the centre $O$ of the base $ABCD$.(a) Find the length of $OA$.(b) Find the angle that edge $VA$ makes with the base.(c) $M$ is the midpoint of $AB$. Find the length $VM$ and hence the angle between the face $VAB$ and the base.