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.
"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).
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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.
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 $A$.
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.
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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.
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.
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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?
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.
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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.
Ramps, regular polygons inscribed in a circle and tangents to a circle from an exterior point are solved by dividing the figure into right-angled triangles.
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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.