Trigonometry

In the diagram, a square is drawn on each side of a right-angled triangle. If the areas of the smaller squares are $9$ $\text{cm}^2$ and $16$ $\text{cm}^2$, what is the area of the largest square?

What is the square of the hypotenuse in a right-angled triangle with legs $7$ and $24$?

Calculate the height of the isosceles triangle shown.

Find the length of $PQ$ in the triangle $PQR$.

A wire of length $25$ m is attached to the top of a $15$ m pole and anchored to the ground. How far is the anchor point from the base of the pole?

Find the value of $x$ for which the triangle with sides $x$, $x+1$, and $\sqrt{2x+1}$ is right-angled, with $x+1$ being the hypotenuse.

A rhombus has diagonals of lengths $16$ cm and $12$ cm. What is the length of one side of the rhombus?

Two circles with radii $5$ cm and $3$ cm have their centers $10$ cm apart. A common external tangent is drawn between them. Calculate the length of this tangent segment.

A point $M$ is the midpoint of side $QR$ of a rectangle $PQRS$. $PQ = 6$ cm and $QR = 8$ cm. Calculate the length of the line segment $PM$.

A circle is inscribed inside a square. If the diagonal of the square is $20$ cm, calculate the radius of the circle to 2 decimal places.

In the diagram, find the angle $x$ to one decimal place.

Calculate the value of $x$ in the right-angled triangle below.

Find the length of side $y$ in the diagram.

What is the length of the hypotenuse in a right triangle where one angle is $45^\circ$ and the opposite side is $7$ cm?

A point $M$ is $10$ units from the origin $O$. If the line $OM$ makes an angle of $35^\circ$ with the $x$-axis, what is the $y$-coordinate of $M$?

Find the length of $x$ in the compound shape.

In the isosceles triangle $ABC$, $AB = AC = 10$ cm and $BC = 12$ cm. Use trigonometry to find the angle $\angle ABC$.

A right-angled triangle has an area of $30$ cm$^2$. If the smallest angle is $25^\circ$, find the length of the hypotenuse to one decimal place.

A rhombus $ABCD$ has diagonals $AC = 16$ cm and $BD = 12$ cm. Calculate the size of the obtuse angle $ABC$ to the nearest degree.

A drone flies from point $S$ at a constant altitude of $120$ m. At one moment, the angle of elevation from an observer $O$ on the ground is $50^\circ$. Two seconds later, the drone is directly over the observer. Calculate the speed of the drone in meters per second.

In triangle $ABC$, $a=7, b=5$ and $\angle C=60^{\circ}$. Calculate the area.

Find the obtuse angle $\theta$ such that $\sin(\theta) = 0.5$.

Using the Sine Rule, find $\angle A$ if $a = 6$, $b = 10$, and $\angle B = 40^{\circ}$.

In $\triangle LMN$, $l = 15$, $m = 20$, and $n = 25$. Determine if the triangle is right-angled by checking $\cos(N)$.

In triangle $PQR$, angle $P = 35^{\circ}$, side $p = 8$ cm, and angle $Q = 72^{\circ}$. Calculate the length of side $q$ to the nearest tenth of a centimeter.

In triangle $ABC$, the side lengths are given as $a = 7$ cm, $b = 10$ cm, and the angle $\angle C = 48^{\circ}$. Calculate the length of side $c$, rounded to one decimal place.

A regular pentagon is inscribed in a circle of radius $10\text{ cm}$. Calculate the length of one side of the pentagon using the Cosine Rule.

In triangle $XYZ$, $XY = 6$ cm and $XZ = 10$ cm. If the area of the triangle is $15$ cm$^2$, find the two possible values for $\angle YXZ$.

A bridge is to be built across a river from point $A$ to point $B$. A surveyor stands at point $C$, $100$ m from $A$. The angle $CAB = 75^{\circ}$ and angle $ACB = 55^{\circ}$. Find the length of the bridge $AB$ to the nearest meter.

A vertical pole $OT$ stands at the center $O$ of a horizontal square $ABCD$. The side length of the square is $10$ m. The angle of elevation of the top of the pole $T$ from corner $A$ is $40^{\circ}$. Calculate the height of the pole $OT$ to two decimal places.

The area of a triangle is $25$ cm$^2$. Two sides are $10$ cm and $10$ cm. What is the angle between them?

Find the area of triangle $ABC$ where $a = 8$, $c = 15$, and $\angle B = 120^{\circ}$.

What is the area of a triangle with $a = 15$ cm, $b = 20$ cm, and $\sin C = 0.8$?

An isosceles triangle has two sides of length $10$ cm and the angle between them is $40^{\circ}$. Find its area to 1 decimal place.

A plot of land is in the shape of a triangle with sides $25$ m and $30$ m. The angle between these sides is $70^{\circ}$. If the land is sold for $100$ per m$^2$, find the total cost.

The area of an isosceles triangle with equal sides of length $s$ and included angle $\theta$ is $A$. If $\theta$ increases from $30^{\circ}$ to $150^{\circ}$, what happens to the area?

In triangle $ABC$, the area is $20$. If $AB=8$ and $\angle A = 45^{\circ}$, find $AC$.

A surveyor measures a triangular field where two sides are in the ratio $2:3$. The angle between them is $40^\circ$. If the area of the field is $500$ m$^2$, find the length of the shorter side to the nearest meter.

A sensor monitors a sector of a circle with a radius of $12$ m. The sector angle is $80^\circ$. A security guard wants to know the area of the triangle formed by the radius lines and the chord connecting their ends. Calculate this area.

The area of an equilateral triangle is $25\sqrt{3}$ cm$^2$. Calculate the side length of the triangle.

A bird flies $30$ m on a bearing of $270^{\circ}$ and then $40$ m on a bearing of $180^{\circ}$. Find the distance from the start.

A point $M$ is $10$ km from $L$ on a bearing of $300^{\circ}$. Calculate the distance of $M$ North of $L$.

Find the bearing of $C$ from $A$ if $B$ is $5$ km East of $A$ and $C$ is $5$ km South of $B$.

A scout walks $4$ km on a bearing of $150^{\circ}$. How far South has the scout walked?

Point $C$ is on a bearing of $210^{\circ}$ from $A$. If the distance $AC$ is $20$ cm, what is the vertical (South) component of the displacement?

A ship leaves port $P$ and sails $10$ km on a bearing of $045^{\circ}$ to point $A$, then $10$ km on a bearing of $135^{\circ}$ to point $B$. Find the bearing of $B$ from $P$.

Town $X$ is $20$ km West of Town $Y$. Town $Z$ is $20$ km South of Town $X$. What is the bearing of $Y$ from $Z$?

Town $B$ is $40$ km from Town $A$ on a bearing of $100^{\circ}$. Town $C$ is $30$ km from Town $A$ on a bearing of $190^{\circ}$. What is the bearing of $B$ from $C$?

From a point $X$, the bearing of $Y$ is $065^{\circ}$. From $Y$, the bearing of $Z$ is $160^{\circ}$. If $XY = 100$ m and $YZ = 150$ m, find the distance $XZ$.

A drone flies $5$ km due North from $O$ to $A$, then $8$ km on a bearing of $120^{\circ}$ to $B$. Find the bearing of $B$ from $O$ to the nearest degree.

In the rectangular cuboid $ABCDEFGH$ shown in the diagram, the dimensions are $AB = 8$ cm, $BC = 6$ cm, and $CG = 5$ cm. Point $G$ is directly above $C$. Calculate the length of the space diagonal $AG$. Round your answer to one decimal place.

Find the length of the altitude of a face of a regular square pyramid if the base side is $8\text{ cm}$ and the vertical height is $3\text{ cm}$.

A right circular cone has a base diameter of $10\text{ cm}$ and a height of $12\text{ cm}$. Find the angle at the vertex of the cone.

A right pyramid has a square base of side $6\text{ cm}$. The slant edges are each $9\text{ cm}$ long. Calculate the vertical height of the pyramid.

A right-angled triangular prism has a base with legs $5\text{ cm}$ and $12\text{ cm}$. The height is $10\text{ cm}$. What is the angle between the longest diagonal of the prism and the $12\text{ cm}$ base edge?

In a rectangular box $ABCD.EFGH$, $AB=8$, $BC=6$, and $AE=5$. What is the angle between the diagonal $BH$ and the edge $BC$?

Calculate the angle between the diagonal of a cube and one of its edges.

A cube has side length $10$ cm. What is the angle between the diagonal of one face and the space diagonal of the cube that share a common vertex?

A cone has a base radius of $5$ cm and a slant height of $13$ cm. A point $P$ is on the circumference of the base. Find the angle between the slant height $VP$ and the diameter through $P$.

A hiker walks $5$ km on a bearing of $060^\circ$ from point $P$ to $Q$. $Q$ is $800$ m higher than $P$. What is the average angle of inclination of the hiker's path?