Trigonometry (DP IB Maths: AI HL)

Adah would like to estimate the height of a tree located at point P on the edge of a riverbank, with the top of the tree at point Q. However, due to a raging river, she is unable to reach the base of the tree. From point M she measures an angle of elevation of 20° to the top of the tree, and then from point N (which is on the edge of Adah’s bank of the river) she measures an angle of elevation of 35° to the top of the tree. Between the points M and N she measures a horizontal distance of 12 m. Points M, N and P all lie on a single horizontal line, and point Q is vertically above point P. The diagram below shows this information.

Calculate the length of NQ.

Calculate the height of the tree. 

Adah borrows a boat and crosses the river at a rate of 50 metres per 15 minutes. 

Assuming that she crosses in a straight line directly from point N to point P, find out how long it takes her to cross the river.

The diagram below shows a triangular field on a farm. AB = 17 m, AC = 45 m and angle = 38°.

X is a point on AC, such that AX :XC is 1 :4.

The field is going to be used for livestock, so a fence is to be installed around its perimeter.

Calculate the total length of fencing required.

The owner of the field had estimated the length of fence required to be 98 m.

Calculate the percentage error in her estimation.

The field is to be divided into two parts by installing a new fence connecting B to X.

Calculate the area of BXC.

The diagram below shows a ship that is located 86 m from an observation station, and a whale that has been spotted from the observation station at a distance of 137 m.

The bearing of the ship from the observation station is  and the whale is located along a bearing of  from the observation station. 

Calculate the distance between the ship and the whale.

Work out the bearing that the ship needs to travel on to reach the whale. Give your answer to 1 decimal place.

The cross-section of a unicorn horn can be modelled by the triangle ABC shown in the diagram below. The length AB = 49 cm and length BC = 58 cm.  The cross-sectional area of the horn is 168 cm².

Find the size of the angle  formed at the tip of horn.

Calculate the length of the base AC that is attached to the unicorn’s head.

The diagram below shows a quadrilateral ABCD.  Angle  =  and angle = . AB = 14.4 cm, AD = 16.2 cm and BC = 19.7 cm.

Calculate the length BD.

Find the size of the angle .

Show that the area of the quadrilateral is 235 cm² correct to the nearest cm².

The diagram below shows the triangular sail of a windsurfing board, ABC, with a horizontal boom PC. AB = 6.1 m and makes an angle of 18° to the vertical. BC = 4.7 m and  = 70°.

Find the area of the whole sail.

Calculate the length of the boom PC.

The area of triangle ABC is .

Calculate the value of

Hence, find BC.

Heron’s formula states that it is actually possible to find the area of any triangle given only its side lengths  and .

The value of s is known as the ‘half perimeter’ where 

Verify that Heron’s formula works for triangle ABC.

The diagram shows a triangular prism ABCDEF of height 18.2 cm.  ED = 7.5 cm, EF = 5.3 cm and AC = 6.6 cm. 

M is the midpoint of BC.

Calculate the length DM.

Find the size of the angle 

Find the area of the triangle EDM.

The diagram below shows a cable-stayed bridge crossing a river from A to B. The height of the embankment at A, measured from the horizontal river bed, is 9.1 m and this drops to a height of 1.3 m at B. The width of the river bed is 90 m.  A vertical central column of height 15 m is situated at the midpoint of the river bed, P, and connects to the exterior supporting cables at point V. The other ends of the cables are attached at points A and B respectively.

Find the size of angle , between the exterior supporting cable and the bridge span.

Calculate the total length of the two exterior supporting cables.

Wynken, Blynken and Nod are three mathematics students.  While the three are revising trigonometry, Nod sets the following problem for his two companions:

“ABC is a triangle with  and . To three significant figures, what is the size of the largest angle in the triangle?”

Wynken and Blynken set to work, and several minutes pass. “70.8 degrees,” states Wynken confidently.  “118 degrees,” insists Blynken a moment later.

Demonstrate that Wynken’s and Blynken’s responses may both be correct answers to the problem Nod has set them.

Suggest an additional piece of information that Nod could provide, that would allow his problem to have a single unique solution.

The diagram below shows a rectangular based pyramid ABCDE. DC = 5.9 cm, AD = 3.7 cm and AE = 7.4 cm. The vertex, E, is positioned directly above the midpoint of the base ABCD.

Find the size of angle 

P is a point located on the edge EB, such that EP : PB = 1 : 4.

Find the area of the triangle EPD.

The diagram below shows a police helicopter using a high beam light at point B to search an area on the ground between A and C. The length of the edge of the light beam that is furthest from the helicopter is 22 m and the angle of depression from the helicopter to the same edge is 47.

The area of the cross section of the search beam, ABC, is 23 m².

Calculate the horizontal length of ground, AC, that is lit by the beam.

Find the size of the angle of the beam 

A spider starts to weave a web, ABCDEFGO, with threads of equal length (AB, BC, etc.) linking the 7 vertices that are equally spaced around the centre point, O. Threads connecting each vertex to the centre (OA, OB, etc.) are also created by the spider. Each line from the centre has a length of 12.6 cm, and the points O, A, B, C, D, E, F and G all lie in a single plane. This is represented in the diagram below.

The spider is located at point G and a fly lands at point D.

Calculate the angle .

The spider decides to add more silk to its web by connecting each vertex to the midpoint of the adjacent line when moving clockwise around the web, for example from G to the midpoint of AO.

Given that the spider can produce 220 cm of silk a day, show that the spider is unable to complete the web on the same day that he started it.

The diagram below shows a funnel in the shape of a right cone with a smaller cone removed from the end. The circular planes at both ends of the object are parallel to one another. The perpendicular height of the complete cone is 168 mm. The initial diameter of the funnel is 98 mm at the upper end and narrows to a diameter of 7 mm at the bottom. 

The funnel has been used to pour some sugar into a bottle and a grain of sugar remains at point P, of the way up the slanted height of the funnel. An ant sits on the edge at the top of the funnel at point A.  is a diameter of the large circular face. Points A, B, P and the axis of the cone all lie on a single plane.

Calculate the direct distance between the ant and the grain of sugar, AP.

Find the size of the angle of depression from the ant to the grain of sugar.

A piece of equipment in a playground consists of two ropes fixed to a hook at point A at the edge of a gap and pulled taut to the other side and fixed at points B and C. The height of the taller left embankment above the ground is 2.3 m. Point B is located at the top of the right embankment. Point C is vertically beneath point B, and the height from point C to the ground is 0.8 m. The angle between the ropes is 23 and the horizontal distance of the gap that is being bridged is 1.4 m. The information is shown on the diagram below. 

Calculate the length BC.

A third piece of rope of length 0.9 m is to be added to the structure at point B and fixed at a point P on the rope AC. 

Calculate the angle  and hence the distance PC.

A symmetrical pendant for a necklace is made in the shape of an irregular hexagon, ABCDEF, as can be seen in the diagram below.  AF, BE and CD are parallel. BE = 22 mm,  AF = CD = 16 mm and = 100. AB = BC and EF = DE. The shaded area is silver and the white area is gold. The width of the entire pendant is 54 mm. This information is shown in the diagram below:

Calculate the percentage of the area of the pendant that is silver.

The ‘H’ on the Hollywood sign is 13.7 m tall, with each leg being 3 m wide and the gap between the legs being 4 m. The width of the cross bar of the ‘H’ (measured from top to bottom in the diagram) is the same as the width of the legs, and the cross bar is situated at the centre of the height of the ‘H’. This information can be seen in the diagram below.

The ‘H’ is situated on horizontal ground but is tilted backwards slightly such that the rear face of the ‘H’ makes an angle of 5  to the vertical. During repair works, a metal support bar is connected between a point A on the ground and point M, which is the midpoint of the rear of the cross bar of the ‘H’. The plane defined by points A, M and B, where B is the midpoint of the gap between the legs of the ‘H’ where they touch the ground, is perpendicular to the plane defined by the rear face of the ‘H’. The angle  between the support bar and the rear face of the ‘H’ is 30°.

Calculate the length of the metal support bar.

Additional support bars are required to reach from the ground to the midpoint at the top of the rear surface of each of the two legs. These additional supports are to contact the ground at the same point as the initial metal support bar. 

Calculate:

A pitched roof is made up of a timber frame, ABCDEF, whose horizontal rectangular base ADFC covers an area of 15.3 m by 8.2 m. The raised central ridge BE is parallel to AD, and its midpoint is situated directly above the intersection of line segments AF and CD. BE is 12.1 m long and is at a height of 2.2 m above the plane defined by ADFC.

Calculate the total length of timber required for the frame.

An internal beam runs from the midpoint of AC, M, to point E.

Calculate the length ME.

In the diagram below, AB, BC and AC are steel beams at the first floor level of a building under construction. ABC lies in a horizontal plane 5 m above ground level, with AB = 7.9 m, BC = 5.8 m and AC = 8.3 m. The line of beam AB makes an angle of 78  with due north when viewed from above.

A pot of paint has been left on beam BC, halfway between points B and C.

Find the bearing from point A to the pot of paint.

A workman is standing on the second floor of the building directly above point A, with his eyes at a height of 12 m above ground level.

Find

from the workman’s eyes to the pot of paint.

A tent with a symmetrical triangular cross-section, ABC, is fixed to a horizontal surface with guy ropes. The tent has a perpendicular height of 1.2 m, and the guy ropes are attached to points X and Y on AB and BC, respectively, such that BX = BY = 0.7 m. The guy ropes are each 1.1 m long, and they are anchored to the ground at points P and Q such that the angles and  are both 30 Points A, B, C, P, Q, X and Y all lie in a single plane. A diagram to illustrate this is provided below. 

Calculate the distance, PQ, between the anchor points of the two guy ropes.

Triangle ABC is such that the length of side AB is  units, the length of side BC is units, and  is an acute angle.

Use a diagram to show that if (and only if)  then there are two triangles,  and , which satisfy the conditions above, where  and are points such that .

Given that , let angle be denoted by ∅.

Write down an expression for ∅ in terms of  and θ.

Write down expressions for angles and  in terms of θ and ∅.

Show that the difference between the areas of triangles  and  is equal to