Trigonometric ratios

Here is a right-angled triangle.

Use trigonometry to work out the value of x.

The diagram shows a square-based pyramid ABCDE. O is the centre of the base.

The length of the base is 10.7 cm. Angle EAO is 67°.

Calculate the length of AE. You must show your working.

The diagram shows two right-angled triangles, ABC and BDC, joined at BC.

Work out angle BCA. You must show your working.

AC is a diagonal of the quadrilateral ABCD.

AD = 72 m. Angle ABC = 108 °, angle BCA = 3 8 ° and angle BAC = 34 °. Angle ACD = 90 °, angle CD A = 65 ° and angle CAD = 25 °.

Find the area of ABCD. You must show your working.

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

The diagram shows a circle, centre O and radius 52 cm.

AB is a chord of the circle. The line through T is a tangent to the circle.

The chord is parallel to the tangent. The perpendicular distance between the chord and the tangent is 72 cm.

Calculate the area of the shaded segment. You must show your working.

The shape below is part of a circle, centre O and radius 12 cm. Angle OAB = 30°.

Work out the perimeter of the shape.

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

A student is using trigonometry to work out an angle, B, in a right-angled triangle. They tell the teacher that sin B = Explain why this student must be wrong.

Here is a right-angled triangle.

Work out the value of x.

The diagram shows a right-angled triangle.

Work out the value of a.

The diagram shows a triangle, ABC, with perpendicular height BD.

BC = 12 cm, angle BCD = 30° and angle BAD = 45°.

Work out the length of BD.

Work out the exact length of AB. Give your answer in its simplest form.

Taylor designs a logo using isosceles triangles joined at a central point, P.

This is the start of Taylor’s design.

The completed design will have rotational symmetry, order 60 about point P.

Each triangle has base, b, and height, h, measured in mm.

Calculate h when b = 40 mm.

Give your answer correct to 1 decimal place.

The diagram shows an isosceles triangle, ABC.

AB = AC = 19 cm. Angle BAC = 54°.

Calculate the length of BC. You must show your working.

The diagram shows two right-angled triangles that are joined together.

All measurements are given accurate to 2 significant figures.

8.3 cm Not to scale

Work out the value of x. Give your answer correct to an appropriate degree of accuracy. You must show your working.

PQS and QRS are triangles.

PQ = 9 cm, QR = 8 cm and RS = 13 cm. Angle QPS = 30° and angle PSQ = 15°.

Calculate angle QSR. Give your answer correct to 1 decimal place. You must show your working.

Here is a right-angled triangle.

Work out the value of x.

The diagram shows triangle ABC. X lies on BC such that angle AXC = 90 °.

BC = 7.5 cm, angle ABC = 32 ° and angle ACB = 43 °.

Work out length AX. You must show your working.

Here is a right-angled triangle.

Work out the value of h.

ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O.

Angle BAC = 70° and BA = BC.

Show that length BO is 17.54 cm, correct to 2 decimal places.

Find the area of triangle ABC. You must show your working.

The diagram shows a quadrilateral ABCD.

AD = 10 cm, angle BAD = 30° and angle ABD = angle BDC = 90°.

The ratio of length BD to length DC is 1: 2.4.

Work out length BC. You must show your working.

Here is a right-angled triangle.

Use trigonometry to work out the value of x.

The diagram shows a circle, centre O.

Points A, B and C lie on the circumference of the circle. Line AOB is a diameter. Line DAE is a tangent to the circle. Angle CAE = 32°.

Give a reason why angle ACB is a right angle.

The radius of the circle is 8 cm.

Calculate length BC.

Here is a right-angled triangle.

Show that angle x is 35°, correct to the nearest degree.

The diagram shows a right-angled triangular prism ABCDEF.

Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm.

Calculate the volume of the prism.

Use trigonometry to show that angle FDC = 31°, correct to the nearest degree.

Calculate the exact length of AF.

Write down the value of sin 45°.

OAB is a sector of a circle, centre O. OA = 6 cm and AX is perpendicular to OB.

The area of sector OAB is 6π cm².

ABC is a right-angled triangle. AB = 20 cm and BC = 37 cm.

Calculate angle BAC.

Here is a right-angled triangle.

Use trigonometry to work out the value of x.

The diagram shows a right-angled triangular prism ABCDEF.

Calculate angle AFB.

In the diagram, ABC is a right-angled triangle. P is a point on AB. BC = 40 m, AP = 20 m and angle ABC = 30 °.

Show that AC = 20 m.

Find the length of PB.

A, B, C and D are points on the circumference of a circle, centre O.

Angle CAD = 28° and CD = 6.4 cm. BD is a diameter of the circle.

Calculate the area of the circle.

The diagram shows two right-angled triangles ABD and BCD, sharing a common side BD. AD = 10 cm, BC = 12 cm and angle DBC = 60°.

Work out the length of AB.

In the diagram below, ABCD is a trapezium. Length AE is 37.5 cm. DE = CF

Find the value of angle x.

The diagram shows a cuboid ABCDEFGH.

Calculate angle BHF.

Alvin has a crate in the shape of a cuboid. The crate is open at the top. The internal dimensions of the crate are 46 cm long by 46 cm wide by 55 cm high.

Alvin has a stick of length 95 cm. Alvin places the stick in the crate so that the shortest possible length extends out above the top of the crate.

Calculate the length of the stick that extends out of the crate.

Calculate the angle the stick makes with the base of the crate.