Trigonometry

Find the value of $k$ if a right-angled triangle has sides $k$, $k+1$, and hypotenuse $\sqrt{25}$.

A television screen is rectangular with a diagonal of 50 inches and a height of 30 inches. What is the width of the screen?

Find the height of an isosceles triangle with base 10 cm and congruent sides of length 13 cm.

The coordinates of two points are $A(1, 2)$ and $B(4, 6)$. Find the distance $AB$.

Find the length of the side $BC$ in the figure, given $AD=5$, $BD=3$, and $AC=13$, with right angles at $D$ and $C$ as indicated by the geometry.

A rhombus has diagonals of length 10 cm and 24 cm. Find the perimeter of the rhombus.

In the triangle shown, $BC$ is perpendicular to $AC$. If $AB = c$, $BC = a$ and $AC = b$, and it is given that $a = x+1$, $b = x-6$ and $c = x+2$. Solve for $x$.

A rope is tied between two poles of heights 10 m and 18 m. The poles are 15 m apart. The rope is pulled taut. If a bird sits on the rope at a horizontal distance of 6 m from the shorter pole, how high is the bird above the ground? (Assume the rope is a straight line).

Point $P$ has coordinates $(3, 4, 12)$ in a 3D Cartesian system. What is the length of the projection of vector $OP$ onto the $xy$-plane, and what is the distance of $P$ from the origin $O$?

A right circular cone has a base radius of 9 cm and a slant height of 15 cm. A smaller cone is cut off from the top, such that the remaining frustum has a height of 4 cm. What is the radius of the top surface of the frustum?

In the right-angled triangle $ABC$ shown below, the side $BC$ has a length of $10$ cm and the angle $\angle ABC$ is $30^\circ$. Calculate the length of the side $AC$.

In the triangle below, if $\sin(\theta) = \frac{1}{2}$, and the hypotenuse is $10$, what is the length of the opposite side $h$?

Based on the graph of $y = \tan(x)$, what is the value of $x$ in the interval $[0, 180^{\circ}]$ where the function is undefined?

In the unit circle shown, the point $Q$ has coordinates $(\cos(210^{\circ}), \sin(210^{\circ}))$. What is the exact value of $\sin(210^{\circ})$?

The diagram shows the graph of $y = \tan(x)$ for $0^\circ \le x \le 180^\circ$. A horizontal line $y = k$ intersects the curve at the point $P(60^\circ, \sqrt{3})$. Find the coordinates of the other point $Q$ in the given range where the curve $y = \tan(x)$ intersects the line $y = -\sqrt{3}$.

In triangle $ABC$, $a = 8, b = 10$ and $\angle A = 40^{\circ}$. Calculate the possible value of $\sin(B)$ using the sine rule.

Given $\cos(\theta) = -\frac{1}{2}$ and $180^{\circ} < \theta < 270^{\circ}$, use the unit circle to find the value of $\sin(\theta)$.

Solve the equation $2\sin^2(x) + 3\cos(x) = 0$ for $0^{\circ} \le x \le 360^{\circ}$.

A regular hexagon is inscribed in a circle of radius $10$ cm. What is the area of the region inside the circle but outside the hexagon?

In the unit circle shown, the point $P$ has coordinates $(\cos \theta, \sin \theta)$. If the area of the shaded region (the segment between the chord and the arc) is $\frac{\pi}{4} - \frac{1}{2}$, find the value of $\theta$ in radians.

In a triangle, the sides are $a=4, b=5, c=6$. What is $\cos C$?

Find the length of the side $BC$ in triangle $ABC$ given $AB=6, AC=10$ and $\angle A = 60^{\circ}$.

In triangle $ABC$, $\angle A = 30^{\circ}, \angle B = 105^{\circ}$ and $c=10$ cm. Calculate side $a$.

In triangle $DEF$, $d=5, e=8$ and $f=7$. Find $\cos E$.

In triangle $ABC$, the sides $a, b, c$ are in arithmetic progression such that $b-a = c-b = d$. If $\cos B = 0.8$, find the ratio $a:b:c$.

In triangle $ABC$, $a=7, b=8, c=9$. Calculate the area of the circumcircle of triangle $ABC$.

In triangle $ABC$, $c^2 = a^2 + b^2 + ab$. Determine the size of angle $C$.

In $\triangle ABC$, $AB=x$, $AC=x+2$ and $\angle BAC = 60^{\circ}$. If the area of the triangle is $10\sqrt{3}$, find the length of $BC$.

A triangle has side lengths $a=7$, $b=8$, and $c=9$. A line is drawn from the vertex opposite side $c$ to the midpoint of side $c$. Calculate the length of this median to one decimal place.

An airplane flies $300$ km on a bearing of $310^{\circ}$ and then $450$ km on a bearing of $070^{\circ}$. How far is the airplane from its starting point?

If the area of a triangle with sides $a=5$ and $b=12$ is $15$, what is the value of $\sin C$?

A triangle has area $20$ square units. If two sides are $10$ and $8$, what is the angle between them in degrees? (Assume the angle is acute)

Find the area of a triangle with sides $7$ cm and $9$ cm and an included angle of $90^\circ$.

In triangle $ABC$, $b = 14$, $c = 10$, and $A = 120^\circ$. Calculate the area.

A right-angled triangle $ABC$ has hypotenuse $AC = 10$ and $\angle BAC = 30^\circ$. Use the sine area formula to verify its area.

A kite $ABCD$ is formed by two triangles $ABD$ and $BCD$ sharing side $BD$. In $\triangle ABD$, $AB=AD=6$ and $\angle BAD=40^\circ$. In $\triangle BCD$, $BC=CD=8$ and $\angle BCD=100^\circ$. Find the total area of the kite to 1 decimal place.

A triangle has area $10$ cm$^2$. One side is $5$ cm and the other is $8$ cm. What is the cosine of the included angle $\theta$ if $\theta$ is obtuse?

The area of a triangle with sides $x$, $x+1$, and an included angle of $60^\circ$ is $\frac{3\sqrt{3}}{2}$. Find the value of $x$.

In a triangle $PQR$, the side $PQ$ is $8$ cm and $PR$ is $x$ cm. The angle $\angle QPR$ is $150^\circ$. If the area of the triangle is $12$ cm$^2$, find the length of the side $QR$.

Triangle $ABC$ has sides $c=7$, $b=8$, and an area of $14\sqrt{3}$. Find the two possible values for side $a$.

Point $Y$ is located $8$ km due East of Point $X$. Point $Z$ is located $6$ km due North of Point $Y$. Find the bearing of $Z$ from $X$ to the nearest degree.

A ship travels from port $P$ for $10$ km on a bearing of $130^\circ$ to point $Q$. Calculate the distance the ship is south of port $P$, to one decimal place.

A hiker walks $5$ km from point $A$ on a bearing of $060^\circ$ to reach point $B$. Which of the following is the correct bearing of $A$ from $B$?

What is the three-figure bearing for the direction South-East?

A boat leaves a dock $D$ and sails $6$ km on a bearing of $300^\circ$ to reach buoy $X$. It then sails $9$ km on a bearing of $210^\circ$ to reach buoy $Y$. Find the bearing of $D$ from $Y$.

Three radio masts $P, Q,$ and $R$ are positioned such that $Q$ is $12$ km from $P$ on a bearing of $080^\circ$. $R$ is $15$ km from $Q$ on a bearing of $170^\circ$. Find the bearing of $R$ from $P$ to the nearest degree.

A ship $S$ is $15$ nautical miles from lighthouse $L_1$ on a bearing of $070^\circ$. The same ship is $20$ nautical miles from lighthouse $L_2$ on a bearing of $160^\circ$. If $L_2$ is due South of $L_1$, find the distance between the two lighthouses $L_1$ and $L_2$.

A helicopter travels from $P$ to $Q$ on a bearing of $150^{\circ}$ and then from $Q$ to $R$ on a bearing of $240^{\circ}$. If $R$ is due south of $P$, and the distance $PQ$ is $100$ km, find the distance $QR$.

A ship $S$ is $20$ nautical miles from lighthouse $L_1$ on a bearing of $300^{\circ}$. The ship is also $25$ nautical miles from lighthouse $L_2$ on a bearing of $040^{\circ}$. If $L_2$ is due east of $L_1$, find the distance between the two lighthouses.

Three landmarks $L_1, L_2,$ and $L_3$ are positioned such that $L_2$ is $5$ km from $L_1$ on a bearing of $160^{\circ}$. $L_3$ is $8$ km from $L_2$ on a bearing of $250^{\circ}$. Find the bearing of $L_1$ from $L_3$.

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

A right circular cone has a base radius of $5$ cm and a slant height of $13$ cm. What is the angle at the vertex (the vertical angle)?

A camera is mounted at $C$, $10$ m above level ground. It tracks an object moving along a straight line $AB$ on the ground. At its closest point $P$, the object is $20$ m from the point $O$ directly below the camera. Find the angle of depression from the camera to $P$.

A hill is modeled as a cone with a circular base of radius $100$ m and height $50$ m. What is the angle of the steepest slope of the hill?

A pyramid has a square base of side $10$ cm. Its height is $12$ cm. Find the angle between two opposite slant edges.

An airplane is flying at an altitude of $3000$ m. From the plane, the angles of depression of two ships in the same vertical plane and on the same side of the airplane are $30^\circ$ and $45^\circ$. Find the distance between the ships.

A surveyor at $P$ measures the angle of elevation of a mountain peak $M$ to be $15^\circ$. After walking $500$ m towards the mountain to point $Q$ on the same horizontal level, the angle of elevation is $20^\circ$. Find the height of the mountain.

A vertical telecommunications mast $VT$ of height $h$ stands at the vertex $V$ of a horizontal triangular base $VAB$. Point $V$ is at the right angle of the triangle $VAB$. It is given that $VA = 30$ m and $VB = 40$ m. The angle of elevation of the top of the mast $T$ from point $A$ is $25^\circ$. Find the angle of elevation of the top of the mast $T$ from the midpoint $M$ of the hypotenuse $AB$.

A square-based pyramid has base side length $10$ m and height $12$ m. Calculate the angle between two adjacent triangular faces.

Two vertical masts $XY$ and $ZW$ of heights $10$ m and $15$ m respectively stand on horizontal ground. The distance between the bases $Y$ and $W$ is $20$ m. A wire is stretched from the top $X$ to the top $Z$. Calculate the angle the wire $XZ$ makes with the horizontal.