Solving right-angled triangles
Picture a wheelchair ramp outside a Luas stop in Dublin, or a surveyor measuring the height of a wind turbine in Co. Kerry without climbing it. Before tackling general triangles with the sine and cosine rules, revise the simplest case: the right-angled triangle. It is the basis of every indirect measurement of heights, distances and slopes, and you will use it in 3D problems too.
Solving a right-angled triangle
"Solving" a triangle means finding all its sides and angles. In a right-angled triangle you already know the right angle: you only need two more pieces of information (two sides, or one side and an acute angle).
A right-angled triangle has legs measuring $5$ cm and $12$ cm. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.
A right-angled triangle has legs measuring $5$ cm and $12$ cm. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.
A ladder $4$ m in length is resting against a vertical wall. If the foot of the ladder is $1.2$ m from the wall, determine the angle the ladder makes with the horizontal…
A ladder $4$ m in length is resting against a vertical wall. If the foot of the ladder is $1.2$ m from the wall, determine the angle the ladder makes with the horizontal ground and the height reached by its top end on the wall.
Points $A(0,0)$, $B(8,0)$, and $C(8,6)$ form a right-angled triangle in the Cartesian plane. Calculate the trigonometric ratios (sine, cosine, and tangent) of the angle…
Points $A(0,0)$, $B(8,0)$, and $C(8,6)$ form a right-angled triangle in the Cartesian plane. Calculate the trigonometric ratios (sine, cosine, and tangent) of the angle at vertex $A$.
Inaccessible heights: angle of elevation
The angle of elevation is the angle between the line of sight to a high point and the horizontal: the classic way of measuring a height without climbing up to it.
An obelisk casts a shadow $15$ m long on the horizontal ground at the moment when the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of…
An obelisk casts a shadow $15$ m long on the horizontal ground at the moment when the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of the obelisk.
A child flies a kite attached to a string of length $45$ m, which is held taut. If the string forms an angle of $55^\circ$ with the horizontal ground, at what height is…
A child flies a kite attached to a string of length $45$ m, which is held taut. If the string forms an angle of $55^\circ$ with the horizontal ground, at what height is the kite? (Assume the child holds the string at ground level).
A straight wheelchair ramp has a length of $6$ m and covers a vertical rise of $0.45$ m. Determine the angle of inclination of the ramp relative to the horizontal.
A straight wheelchair ramp has a length of $6$ m and covers a vertical rise of $0.45$ m. Determine the angle of inclination of the ramp relative to the horizontal.
A rectangular solar panel $2.5$ m long is inclined at $35^\circ$ to the horizontal on a flat roof. Calculate the length of the shadow cast by the panel on the roof when…
A rectangular solar panel $2.5$ m long is inclined at $35^\circ$ to the horizontal on a flat roof. Calculate the length of the shadow cast by the panel on the roof when the sun's rays strike completely perpendicular to the plane of the roof.
Angle of depression
Looking down from a high point, the angle between the line of sight and the horizontal is the angle of depression. By alternate angles between parallel lines, it is equal to the angle of elevation seen from the low point.
To calculate the distance between the base of a coastal lighthouse $24\text{ m}$ high above sea level and an anchored boat, Lucas measures an angle of depression of…
To calculate the distance between the base of a coastal lighthouse $24\text{ m}$ high above sea level and an anchored boat, Lucas measures an angle of depression of $20^\circ$ from the lighthouse lantern. In his notebook, he writes:
$$\tan(20^\circ) = \frac{x}{24} \implies x = 24 \times \tan(20^\circ) \approx 8.74\text{ m}$$- a) Without using a calculator, explain through elementary geometric reasoning (mentally comparing with the shape of a $45^\circ$ triangle) why a result of $8.74\text{ m}$ is clearly absurd for an angle of depression of only $20^\circ$.
- b) Identify the conceptual error Lucas made by placing the angle of depression with respect to the vertical instead of the horizontal.
- c) Correct the formulation and calculate the actual horizontal distance $x$ between the lighthouse and the boat.
From the lantern room of a lighthouse, located $45$ m above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression to…
From the lantern room of a lighthouse, located $45$ m above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression to the nearer ship is $34^\circ$, and to the farther ship is $18^\circ$. Calculate the exact distance separating the two ships.
Two vertical industrial chimneys of heights $20$ m and $35$ m are built on flat ground. From the midpoint of the line segment joining their bases, the top of the first…
Two vertical industrial chimneys of heights $20$ m and $35$ m are built on flat ground. From the midpoint of the line segment joining their bases, the top of the first chimney is observed with an angle of elevation of $35^\circ$. What will be the angle of elevation of the top of the second chimney from the same midpoint?
Distances from two points: a system of equations
When you cannot measure a height directly, another strategy is to take the angle of elevation from two different points in line with the base, and set up a system with the same height $h$ in both equations.
From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. Moving $25$ m in a straight line towards the base of the…
From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. Moving $25$ m in a straight line towards the base of the tower, the new angle of elevation is $50^\circ$. Set up a system of trigonometric equations and calculate the height of the wind turbine.
Two observers, $A$ and $B$, are separated by a distance of $300$ m on a horizontal plain. Both look towards a stationary drone that is located exactly in the vertical…
Two observers, $A$ and $B$, are separated by a distance of $300$ m on a horizontal plain. Both look towards a stationary drone that is located exactly in the vertical plane passing through the straight line joining them, positioned between them. Observer $A$ sees the drone with an angle of elevation of $60^\circ$ and observer $B$ sees it with an angle of $45^\circ$. Calculate the exact altitude at which the drone is flying, expressing the result in surd form.
Combined problems: ramps, polygons and tangents
Ramps, regular polygons inscribed in a circle, tangents to a circle from an exterior point and solids in 3D are solved by finding right-angled triangles inside the figure.
Common chords and perpendicular bisectors
The perpendicular from the centre of a circle to a chord bisects the chord (Theorem 18 in the syllabus). This gives a right-angled triangle with the radius as hypotenuse and half the chord as one leg. For two circles that intersect, the line joining the centres bisects the common chord at right angles.
Two intersecting circles have radii of $10$ cm and $17$ cm. The length of the common chord of both circles is $16$ cm, and the centres lie on opposite sides of this…
Two intersecting circles have radii of $10$ cm and $17$ cm. The length of the common chord of both circles is $16$ cm, and the centres lie on opposite sides of this chord. Calculate the distance between the centres of the two circles and determine the angle formed between their respective tangent lines at one of the points of intersection.
Bearings: directions measured from North
A bearing is the direction of travel or of an object, measured as an angle from North. When a problem says "the angle relative to North", draw a North line through the observer, then build a right-angled triangle from the east-west and north-south displacements.
A coastguard detects that a ship $A$ is sailing in a straight line towards the East at a constant speed of $20$ knots, and a ship $B$ is sailing towards the North at…
A coastguard detects that a ship $A$ is sailing in a straight line towards the East at a constant speed of $20$ knots, and a ship $B$ is sailing towards the North at $15$ knots. At a given moment ($t=0$), both ships are at exactly the same point, where their paths cross. At what angle relative to North will the captain of ship $A$ observe ship $B$ exactly $2$ hours after crossing?
Inscribed rectangles and similar triangles
When a rectangle sits inside a triangle with one side on the base, the small triangle above the rectangle is similar to the whole triangle (the top side is parallel to the base). Use the ratio of heights to the ratio of bases, or use the tangent of a base angle of the large triangle.
In an isosceles triangle whose base is $12$ cm long and whose altitude above this base is $18$ cm, a rectangle is inscribed such that one of its sides lies on the base…
In an isosceles triangle whose base is $12$ cm long and whose altitude above this base is $18$ cm, a rectangle is inscribed such that one of its sides lies on the base of the triangle and the other two vertices touch the equal sides. If the diagonals of this rectangle intersect forming an angle of $60^\circ$ (this being the angle subtending the base of the rectangle), set up the trigonometric and geometric system required to calculate the exact dimensions of this rectangle.
Right-angled triangles in 3D
In 3D, draw a right-angled triangle in a flat plane first, then use its result in a second right-angled triangle.
An isosceles triangle has two equal sides measuring $14$ cm each, and the non-equal angle (the included angle between them) measures $40^\circ$. By drawing the altitude…
An isosceles triangle has two equal sides measuring $14$ cm each, and the non-equal angle (the included angle between them) measures $40^\circ$. By drawing the altitude to the base to divide it into two right-angled triangles, calculate the length of that base.
Calculate the area and perimetre of a regular hexagon inscribed in a circle of radius $12$ cm. (Remember to divide the polygon into triangles to use trigonometry).
Calculate the area and perimetre of a regular hexagon inscribed in a circle of radius $12$ cm. (Remember to divide the polygon into triangles to use trigonometry).
A road sign warns of a steep gradient of $15\%$. This indicates that for every $100$ m travelled in the horizontal projection, there is a vertical rise of $15$ m.…
A road sign warns of a steep gradient of $15\%$. This indicates that for every $100$ m travelled in the horizontal projection, there is a vertical rise of $15$ m. Calculate the angle of inclination of the road and the actual distance a vehicle will travel on the road surface if it advances $600$ m in the horizontal projection.
A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, situated $45$ m from the base of the building, a surveyor observes…
A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, situated $45$ m from the base of the building, a surveyor observes the base of the antenna with an angle of elevation of $38^\circ$ and the top of the antenna with an angle of elevation of $46^\circ$. Calculate the height of the building and the exact length of the antenna.