Geometry and Trigonometry

The angle of elevation from a point $G$ on the ground to the top of a tower $T$ is $30^\circ$. The tower is 20m high. How far is $G$ from the base of the tower $B$?

A point $M$ is the midpoint of the side $BC$ of a rectangle $ABCD$ in the $xy$-plane. $A(0,0,0)$, $B(6,0,0)$, $C(6,4,0)$, $D(0,4,0)$. A vertical pole $VP$ is at $A$ with height 3. Find the distance from $V(0,0,3)$ to $M$.

Calculate the size of the angle between the diagonal of a face and the edge of a cube.

A sphere is inscribed exactly inside a cube with side length 10 cm. What is the distance from the center of the sphere to any corner of the cube?

A sphere of radius $R$ fits perfectly inside a cylinder of the same radius $R$ and height $2R$. What fraction of the cylinder's volume is occupied by the sphere?

A wedge has a rectangular base of $10 \times 6$ units. The vertical triangular face has a height of 4 units and is perpendicular to the base along the 6-unit side. Find the length of the longest edge of the wedge.

The angle between a line and a plane is defined as the angle between the line and its projection on the plane. A cube has side length 1. Find the sine of the angle between the space diagonal and the base plane.

A right circular cone is inscribed in a sphere of radius $R$. If the height of the cone is $h$, find the radius $r$ of the cone's base in terms of $R$ and $h$.

The angle between a space diagonal of a cube and one of its faces is $\theta$. Find the value of $\cos^2 \theta$.

Point $P(x, y, z)$ is $13$ units from the origin. If $P$ lies on the line where $x=3$ and $y=4$, determine the possible values of $z$.

In triangle $ABC$, $c = 8$, $\angle A = 45^{\circ}$, and $\angle C = 30^{\circ}$. Find the length of side $a$.

A triangle has sides $4, 5, 6$. Find the cosine of the largest angle.

In triangle $ABC$, $\sin A = \frac{1}{3}$, $a = 6$, and $b = 9$. Find $\sin B$.

Triangle $ABC$ has $a = 10$, $b = 10$, and $c = 10\sqrt{2}$. What is the measure of angle $C$?

A triangle has an area of $6$, and two of its sides are $3$ and $8$. Find the possible values of the third side.

In triangle $ABC$, $BC = \sqrt{2}$, $AC = 2$, and $\angle ABC = 135^{\circ}$. Find the measure of $\angle BAC$.

In a triangle, the sides are in the ratio $3:5:7$. Find the measure of the largest angle.

A triangle $ABC$ has sides $AB = 6$, $BC = 10$, and $AC = 14$. Show that one of the angles is $120^{\circ}$ and calculate the area of the triangle.

A aircraft $A$ is $10$ km from a control tower $O$ on a bearing of $060^{\circ}$. Another aircraft $B$ is $15$ km from $O$ on a bearing of $150^{\circ}$. Calculate the distance between aircraft $A$ and $B$.

In $\triangle ABC$, $BC = a$, $AC = b$, and $AB = c$. If $a^2 = b^2 + c^2 - bc$, find the measure of angle $A$ in degrees.

Calculate the length of the diagonal of a rectangular box with dimensions $3, 4, 12$.

A ship travels $d\text{ km}$ on a bearing of $\theta$. It then turns and travels the same distance $d$ on a bearing of $180+\theta$. What is its final position relative to the start?

A point $M$ is the midpoint of the side $AB$ of a square $ABCD$ with side length $2$. Calculate the distance from $D$ to $M$.

Find the angle between the diagonal of a face and the side of a cube.

A ship leaves port $P$ and sails $12$ km on a bearing of $040^\circ$ to reach a buoy $B$. A lighthouse $L$ is located $15$ km due East of port $P$. Calculate the distance from the buoy $B$ to the lighthouse $L$.

An airplane takes off from a point $A$ and flies on a bearing of $060^\circ$ for $400$ km to point $B$. It then changes course and flies on a bearing of $150^\circ$ for $300$ km to point $C$. Find the direct distance $AC$ in kilometers.

A flagpole $TF$ of height $10\text{ m}$ stands on a flat roof. From a point $A$ on the ground, the angle of elevation of the top $T$ is $45^\circ$ and the angle of elevation of the foot $F$ is $30^\circ$. Find the height of the building.

A cube $ABCDEFGH$ has side length $10$. Point $M$ is the midpoint of $GH$. Find the cosine of the angle between the line segment $AM$ and the base $ABCD$.

Two points $P$ and $Q$ are on a level plain. A balloon is at $B$, directly above a point $R$ which lies on the line segment $PQ$. The angle of elevation of $B$ from $P$ is $35^\circ$ and from $Q$ is $42^\circ$. If $PQ = 100$ m, find the height of the balloon.

A pyramid has a rectangular base $ABCD$ where $AB = 8$ cm and $BC = 6$ cm. The vertex $V$ is directly above the center of the base. If the volume is $96$ $\text{cm}^3$, find the angle that the edge $VA$ makes with the base.

Determine the area of a sector if its arc length is $10 \text{ cm}$ and its radius is $5 \text{ cm}$.

A pendulum of length $80 \text{ cm}$ swings through an angle of $0.2$ radians. Find the distance traveled by the tip of the pendulum in one swing.

On the unit circle, point $Q$ corresponds to the angle $\theta = -\frac{\pi}{6}$. What are the coordinates of $Q$?

The arc length of a circle's sector is $5\pi \text{ cm}$. If the radius is $15 \text{ cm}$, find the central angle in degrees.

If $\sin\theta = \frac{1}{3}$ and $\theta$ is in the second quadrant, find the exact value of $\cos\theta$.

The perimeter of a sector is exactly half the circumference of the circle. Find the central angle $\theta$ in radians.

A point $P(x, y)$ on the unit circle has $x = -0.6$ and is in the third quadrant. Find the value of $\tan\theta$.

In a circle of radius $R$, an arc of length $L$ subtends an angle $\theta$ at the center. A second circle is drawn with a radius equal to the chord length of this arc. Find the ratio of the area of the second circle to the area of the first circle in terms of $\theta$.

Two circles of radius $10$ cm intersect such that each passes through the center of the other. Calculate the area of the region common to both circles.

A running track consists of two parallel straight sections of length $L$ and two semicircular ends of radius $r$. The total distance around the inside of the track is $400$ m. If the area enclosed by the track is to be maximized, find the value of $r$ in terms of $L$.

The diagram shows a unit circle centered at the origin $O(0,0)$. A point $P$ lies on the circle such that the angle $\theta$ is formed with the positive $x$-axis. If the coordinates of point $P$ are $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$, find the value of $\tan \theta$.

If $\tan \theta = 0$, what is a possible value for $\theta$ in the interval $[0, 2\pi)$?

Which of the following points lies on the unit circle?

Find the value of $\cos \pi$.

Which of the following is equivalent to $\sin(\pi - \theta)$?

If $\cos \theta = -\frac{12}{13}$ and $\pi < \theta < \frac{3\pi}{2}$, find the value of $\tan \theta$.

In the unit circle, if $\sin \theta = a$ and $\cos \theta = b$, find the value of $a^2 + b^2$.

A Ferris wheel is represented by a unit circle in the $xy$-plane. A rider's position is $(x, y)$. If the rider has rotated by an angle $\theta$ from $(1, 0)$ such that $\sin \theta = \cos \theta$ and $\pi < \theta < 2\pi$, find the value of $\theta$.

In a geometry proof, a student uses the unit circle to show the relationship between angles. If $\theta$ is an acute angle such that $\cos \theta = p$, express $\tan(\pi - \theta)$ in terms of $p$.

A rotating beacon is modeled on a unit circle. The beacon's light is at point $L(x, y)$. If the angle $\theta$ satisfies $\tan \theta = -\frac{12}{5}$ and $\cos \theta > 0$, calculate the $y$-coordinate of $L$.

State the range of the function $f(x) = 5\sin x - 2$.

The identity $\tan x \cos x$ simplifies to:

Find the value of $x$ in the interval $0 \le x \le \frac{\pi}{2}$ that satisfies $\cos x = \frac{\sqrt{2}}{2}$.

Identify the amplitude of the function shown in the graph.

A triangle has sides of length $5$ cm and $8$ cm, and the angle between them is $60^\circ$. Find the area of the triangle.

Find all solutions to $\tan x = \sqrt{3}$ for $0 \le x \le 2\pi$.

Express $3\sin^2 x + 3\cos^2 x + 4\cos(2x)$ in terms of $\cos x$.

A coastal town is monitoring the tide level in its harbor. The height of the water, $H$ metres relative to a mean sea level, is modeled by the function $H(t) = 3\sin(kt) - 4\cos(kt)$, where $t$ is the time in hours after midnight and $k = \frac{\pi}{6}$. A local fisherman needs to know the maximum height the tide reaches and at what time after midnight this first occurs. By expressing $H(t)$ in the form $A\sin(kt - \alpha)$, where $A > 0$ and $0 < \alpha < \pi$, determine the maximum height and the first time $t > 0$ (in hours) that it is achieved.

A hiker walks $5$ km from point $A$ on a bearing of $070^\circ$ to reach point $B$. They then walk $8$ km on a bearing of $160^\circ$ to reach point $C$. Calculate the distance from point $A$ to point $C$.

Solve the equation $2\sin^2 x + 3\cos x - 3 = 0$ for $0 \le x \le 2\pi$.

Find the value of $k$ if the function $f(x) = \sin(x) + k$ passes through the point $(0, 5)$.

The diagram shows a sector with radius $r=6$ and arc length $s=3$. Find the angle $\theta$ in radians.

Which value corresponds to $\tan(45^\circ)$ on the unit circle?

Find the area of the triangle shown in the diagram.

The identity $\cos(2x) = 1 - 2\sin^2(x)$ is used to simplify $f(x) = 10 \sin^2(x)$. Which of the following is an equivalent expression for $f(x)$?

Given $\tan(x) = 2$ and $x$ is in the third quadrant, find the value of $\sin(x)$.

Solve $\cos^2(x) - \sin^2(x) = 0$ for $0 \le x \le \pi$.

In triangle $PQR$, $PQ = 10$, $QR = 12$, and the area of the triangle is $30\sqrt{3}$. Find the two possible values for the measure of angle $PQR$.

A mountain climber at point $A$ observes the peak of a mountain $P$ at an angle of elevation of $30^{\circ}$. After walking $800$ meters directly towards the mountain base on level ground to point $B$, the angle of elevation is $45^{\circ}$. Calculate the height of the mountain $h$ in meters.

Given that $\sin \theta = \frac{3}{5}$ and $\frac{\pi}{2} < \theta < \pi$, find the exact value of $\cos(2\theta)$.

Identify the identity represented by the graph $y = \cos^2 x - \sin^2 x$.

Using the unit circle shown, what is the value of $\tan(\pi - \theta)$ in terms of $t = \tan \theta$?

If $\sin \theta = p$, express $\cos(2\theta)$ in terms of $p$.

Simplify $\cos(2x)$ when $\cos x = \frac{1}{\sqrt{3}}$ using the identity $\cos(2x) = 2\cos^2 x - 1$.

In the diagram, a ladder makes an angle $\theta$ with the ground. If the ladder is $L$ meters long, the height it reaches is $h = L \sin \theta$. If we double the angle to $2\theta$, what is the new height $H$ in terms of $L, \sin \theta$, and $\cos \theta$?