Right angle triangles (SOH CAH TOA): notes and practice questions

A ladder must be placed against a tall vertical building, clearing a monument that is 8 m high and stands on horizontal ground 1 m away from the building's base. The ladder touches the ground, the top corner of the monument, and the wall of the building.

Let $L$ be the length of the ladder in metres.

Let $\theta$ be the angle that the ladder makes with the ground, where $0 < \theta < \frac{\pi}{2}$.

(a) Show that $L = \sec \theta + 8\csc \theta$.

(b) (i) Find $\frac{dL}{d\theta}$.

(b) (ii) When $\frac{dL}{d\theta} = 0$, show that $\tan\theta = 2$.

(c) (i) Find $\frac{d^2L}{d\theta^2}$.

(c) (ii) When $\tan\theta = 2$, find the value of $\frac{d^2L}{d\theta^2}$.

(d) (i) Hence, justify that $L$ is a minimum when $\tan\theta = 2$.

(d) (ii) Determine this minimum value of $L$.

(e) A construction company only has ladders with a maximum length of 11 m. Determine whether it is possible to position a ladder against the building over the monument, giving a reason for your answer.

(a) A new Ferris wheel is being constructed. From a point A on the ground, 15 metres from the base of the support tower, the angle of elevation to the centre C of the Ferris wheel is 33.7 degrees. Find the height of point C above the ground.

(b) An engineer walks 5 metres closer to the support tower to point B. Find the angle of elevation of point C from point B, giving your answer in radians.

(c) The lowest point a passenger cabin reaches is 1.5 metres above the ground, allowing for easy boarding. Calculate the radius of the Ferris wheel.

(d) The height $h$, in metres, of a passenger cabin above the ground can be modelled by the function $h(t) = q - p \cos(kt)$, where $t$ is the time in seconds after the cabin starts moving from its lowest point. The Ferris wheel completes one full rotation in 40 seconds.

Find the values of $p$, $q$, and $k$.

(e) An observer watches the Ferris wheel for 5 minutes. How many times does a specific cabin pass its highest point during this time?

(a) A surveyor is mapping a new development. From point A, the distance to a prominent tree (T) is measured as 300 m. From another point B, 250 m away from A, the distance to the same tree (T) is measured as 200 m.

Find the measure of angle ATB.

(b) Find the shortest distance from the tree (T) to the line segment AB.

(a) Calculate the shortest distance from the apex $V$ to the side $AB$ of the pyramid's base.

(b) Find the total surface area of the four triangular glass panels forming the pyramid's roof.

(c) Calculate the total height of the monument from the ground to the apex $V$.

(d) Determine the size of the angle between the slant edge $VA$ and the base edge $AB$.

(e) A security camera is installed at a point $S$ on the vertical edge $C'C$ of the monument. A guard standing at point $B'$ (a corner of the base of the monument) measures the angle of elevation to point $S$ to be $40^\circ$. Find the height of the security camera from the ground.

(a) A surveyor is measuring a tall communications mast. From a point $P$ on the ground, the angle of elevation to the top of the mast is $38^\circ$. The total height of the mast from the ground to its top is $70$ meters. Calculate the horizontal distance $d$ from point $P$ to the near edge of the mast's base, giving your answer correct to one decimal place.

(b) Calculate the direct line-of-sight distance $L$ from point $P$ to the top of the mast, giving your answer correct to one decimal place.

(c) A maintenance platform is located $15$ m below the top of the mast. Find the angle of depression from this maintenance platform to point $P$, giving your answer in degrees correct to one decimal place.

A designer is creating a symmetrical display stand for a museum exhibit. The cross-section of the stand is an isosceles trapezium $ABCD$, where $BC$ is parallel to $AD$. The shorter parallel side $BC$ measures $15$ cm, the longer parallel side $AD$ measures $27$ cm, and the non-parallel sides $AB$ and $CD$ each measure $10$ cm.

Show that the height of the trapezium is $8$ cm.

Hence, find the area of the cross-section of the display stand.

Find the size of the angle $ADC$, giving your answer to one decimal place.

Calculate the length of the diagonal $AC$, giving your answer to three significant figures.

(a) The main antenna of a communication tower is located at point $A(2, 8, 10)$. The centre of the rectangular base of the tower is at point $C(0, 5, 0)$. A vertical support beam connects the antenna to the centre of the base. Calculate the length of this support beam, $AC$.

(b) The rectangular base of the tower has dimensions $6$ m by $4$ m. Calculate the length of the diagonal of this base.

(c) A support cable runs from the antenna $A$ to one of the corners of the base, say point $P$. Find the size of the angle that this support cable $AP$ makes with the base platform.

A company produces solid glass paperweights. One model is a right pyramid with a square base of side length 10 cm and a height of 12 cm.

Find the volume of one of these paperweights.

Find the slant height of the pyramid.

The company is investigating a new design for a paperweight, a cuboid with the same square base and the same total surface area as the pyramid.

Show that the total surface area of the pyramid is 360 cm$^2$.

Find the height, $H$, of the cuboid-shaped paperweight.

The company wants to choose the design that uses less glass in order to reduce production costs.

State whether they should switch to the cuboid design. Justify your conclusion.