Exercises: Solving right-angled triangles

A right-angled triangle has legs measuring $5$ in and $12$ in. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.

A ladder of length $4$ ft is leaning against a vertical wall. If the base of the ladder is $1.2$ ft from the wall, determine the angle the ladder makes with the horizontal ground and the height its top end reaches on the wall.

An obelisk casts a shadow $15$ ft long on the horizontal ground at the moment the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of the obelisk.

A child is flying a kite held by a string of length $45$ ft, which is kept completely 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).

In a rectangular computer screen, the diagonal measures $60$ in and forms an angle of $29^\circ$ with the bottom base of the frame. Calculate the width and height of the screen.

A straight wheelchair access ramp has a length of $6$ ft and overcomes a vertical drop of $0.45$ ft. Determine the angle of inclination of the ramp with respect to the horizontal.

An isosceles triangle has two equal sides measuring $14$ in each, and the unequal angle (the one between them) measures $40^\circ$. By drawing the height corresponding to the base to divide it into two right-angled triangles, calculate the length of this base.

The points $A(0,0)$, $B(8,0)$ and $C(8,6)$ form a right-angled triangle on the Cartesian plane. Calculate the trigonometric ratios (sine, cosine and tangent) of the angle located at the vertex $A$.

From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. After advancing $25$ ft 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.

Calculate the area and the perimeter of a regular hexagon inscribed in a circle of radius $12$ in. (Remember to divide the polygon into triangles to use trigonometry).

From the lantern of a lighthouse, located $45$ ft above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression at which the closer ship is seen is $34^\circ$, and that of the further ship is $18^\circ$. Calculate the exact distance separating the two ships.

The diagonals of a rhombus measure $24$ in and $10$ in. Knowing that the diagonals of a rhombus intersect perpendicularly at their midpoint, calculate the measure of the sides of the rhombus and the size of its four interior angles.

A traffic sign warns of a prolonged gradient of $15\%$. This indicates that for every $100$ ft advanced in the horizontal projection, there is a vertical ascent of $15$ ft. Calculate the angle of inclination of the road and the actual distance a vehicle will travel on the road surface if it advances $600$ ft in the horizontal projection.

Two vertical industrial chimneys of heights $20$ ft and $35$ ft are built on flat ground. From the midpoint of the straight line segment joining their bases, the top of the first chimney is observed with an angle of elevation of $35^\circ$. With what angle of elevation will the top of the second chimney be observed from that same midpoint?

From a point $P$ outside a circle of radius $8$ in, the two tangent lines to it are drawn. If the distance from the point $P$ to the center of the circle is $17$ in, calculate the total angle formed between the two tangent lines.

A rectangular solar panel $2.5$ ft long is inclined at $35^\circ$ to the horizontal of a flat roof. Calculate the length of the shadow the panel casts on the roof when the sun's rays strike completely perpendicular to the plane of the roof.

A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, located $45$ ft 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.

In a right-angled triangle, the length of one leg is $x$ in, that of the other leg is $(x+7)$ in and the hypotenuse measures $(x+14)$ in. Set up and solve the corresponding equation to determine the value of $x$. Then, calculate the exact value (as an irreducible fraction) of the sine and cosine of the smallest angle of the triangle.

Two observers, $A$ and $B$, are separated by a distance of $300$ ft on a horizontal plain. Both look towards a stationary drone located exactly in the vertical plane passing through the straight line joining them, and situated spatially 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 height at which the drone is flying, expressing the result with radicals.

Two intersecting circles have radii of $10$ in and $17$ in. The length of the common chord to both circles is $16$ in, and the centers lie on opposite sides of this chord. Calculate the distance separating the centers of the two circles and determine the angle formed between their respective tangent lines at one of the points of intersection.

A straight ramp is to be built to overcome a vertical drop of height $h$. If the angle of inclination of the ramp with respect to the horizontal is $\alpha$, prove using the definitions of the trigonometric ratios that the sum of the length of the base of the ramp (its horizontal projection) plus the length of the ramp itself is equal to $h \cdot \frac{1 + \cos(\alpha)}{\sin(\alpha)}$.

A Coast Guard 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 instant ($t=0$), both ships are at exactly the same point, where their paths cross. At what angle with respect to the North direction will the captain of ship $A$ observe ship $B$ exactly $2$ hours after crossing?

In an isosceles triangle whose base measures $12$ in and whose height above this base is $18$ in, a rectangle is inscribed so that one of its sides rests 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 necessary to calculate the exact dimensions of this rectangle.

A vertical pole casts a shadow on flat ground. At a certain time in the morning, when the sun's elevation is $\alpha = 30^\circ$, the shadow measures $L$ feet. A couple of hours later, when the sun's elevation has increased to $\beta = 45^\circ$, it is found that the shadow has shortened by exactly $10$ feet. Set up a system of equations and calculate the exact height of the pole expressed with radicals.