You already know what sine, cosine and tangent are. Now it's time to use them for what really matters: finding measurements that you can't take directly, such as the height of a building, the width of a river or the slope of a ramp.
"Solving" a right-angled triangle means calculating all of its sides and all of its angles. Since it already has a right angle, you only need two more pieces of information (two sides, or one side and an acute angle) to find everything else.
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Solve a right-angled triangle in which one side measures $7$ cm and its opposite angle is $40^\circ$.
Solve a right-angled triangle in which one side measures $7$ cm and its opposite angle is $40^\circ$.
Going further: Inaccessible heights: the angle of elevation
The angle of elevation is the angle that the line of sight to a high point makes with the horizontal. It is the classic tool for measuring the height of something without climbing it: a building, a tree, a tower.
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From a point on the ground located $10$ m from the base of a tree, its top is observed with an angle of elevation of $30^\circ$. How tall is the tree?
From a point on the ground located $10$ m from the base of a tree, its top is observed with an angle of elevation of $30^\circ$.
From a point on the ground $25$ m from the facade of a building, the roof is seen with an angle of elevation of $50^\circ$. What is the height of the building?
From a point on the ground $25$ m from the facade of a building, the roof is seen with an angle of elevation of $50^\circ$. What is the height of the building?
Going further: The angle of depression: looking downwards
When you look down from a high point (a cliff, a lighthouse, a plane) at an object on the ground, the angle that your line of sight makes with the horizontal is called the angle of depression. By geometry (they are alternate angles between parallel lines), the angle of depression from above is equal to the angle of elevation seen from below.
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From the top of a $60$ m cliff, a buoy is sighted with an angle of depression of $25^\circ$, as shown in the figure. a) How far from the foot of the cliff is the buoy?…
From the top of a $60$ m cliff, a buoy is sighted with an angle of depression of $25^\circ$, as shown in the figure.
a) How far from the foot of the cliff is the buoy? b) How far is it from the observer?
From the top of a $45$ m lighthouse, two ships are observed in line with the lighthouse: the nearer one with an angle of depression of $32^\circ$ and the farther one…
From the top of a $45$ m lighthouse, two ships are observed in line with the lighthouse: the nearer one with an angle of depression of $32^\circ$ and the farther one with $20^\circ$, as shown in the figure.
When you can't measure a distance directly (the width of a river, the height of a tower whose base you can't reach), a very powerful strategy is to take two measurements from two different points and combine the information.
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To measure the width of a river, you stand directly opposite a tree on the other bank and walk $30$ m along your bank. From there, the tree is seen forming an angle of…
To measure the width of a river, you stand directly opposite a tree on the other bank and walk $30$ m along your bank. From there, the tree is seen forming an angle of $55^\circ$ with the bank, as shown in the figure.
From a certain point in a valley, the summit of a mountain is seen with an angle of elevation of $35^\circ$. Walking back $250$ m in a straight line, the summit is seen…
From a certain point in a valley, the summit of a mountain is seen with an angle of elevation of $35^\circ$. Walking back $250$ m in a straight line, the summit is seen with an angle of $22^\circ$, as shown in the figure.
a) Set up a system of two equations using the tangent in each of the two right-angled triangles. b) Solve the system. c) What is the height of the mountain above the observer's eye level?
A group of students wants to measure the width of a lake without crossing it. Opposite a buoy $B$ located on the far bank, they mark a point $P$ on their bank; from $P$…
A group of students wants to measure the width of a lake without crossing it. Opposite a buoy $B$ located on the far bank, they mark a point $P$ on their bank; from $P$ they walk $60$ m along the bank in a perpendicular direction to a point $Q$, and from $Q$ they measure the angle $\widehat{PQB} = 48^\circ$. a) Draw a diagram of the situation. b) Calculate the width of the lake. c) If a straight footbridge were built from $P$ to $B$ with an additional $5\ \%$ in length for the supports, what length would the footbridge be?
You can do it on paper and send a photo with your answer.
Ramps, slides, roofs and mountain passes are described by their angle of inclination and their length: two pieces of information about a right-angled triangle in which the hypotenuse is the ramp, the base is the horizontal distance and the height is the rise. Some problems combine two triangles at once.
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A water park slide has a $18$ m ramp that forms an angle of $25^\circ$ with the ground, as shown in the figure. a) How high is the starting platform? b) What horizontal…
A water park slide has a $18$ m ramp that forms an angle of $25^\circ$ with the ground, as shown in the figure.
a) How high is the starting platform? b) What horizontal distance does the slide occupy?
The ramp at a skatepark has an incline of $18^\circ$ and a length of $6$ m, as shown in the figure. a) What vertical height does the ramp cover? b) What is its gradient…
The ramp at a skatepark has an incline of $18^\circ$ and a length of $6$ m, as shown in the figure.
a) What vertical height does the ramp cover? b) What is its gradient expressed as a percentage?
A zip wire connects two towers that are $22$ m and $14$ m tall, separated horizontally by $60$ m, as shown in the figure. a) What is the difference in height between the…
A zip wire connects two towers that are $22$ m and $14$ m tall, separated horizontally by $60$ m, as shown in the figure.
a) What is the difference in height between the two ends? b) What is the length of the cable, assuming it is taut and straight? c) At what angle of descent does the user slide?