Trigonometry

In a right-angled triangle, the legs are $x$ and $2x$. The hypotenuse is $10$. Find $x$.

Calculate the length of the diagonal of a rectangle with dimensions $15$ cm by $20$ cm.

A ship travels $20$ km North and then $21$ km East. What is the direct distance from the starting point?

In a right triangle, the squares of the legs are $49$ and $576$. What is the hypotenuse?

A right-angled triangle has sides in arithmetic progression. If the shortest side is $3k$, what is the length of the hypotenuse?

A point $P(x, y)$ lies on a circle of radius $r$ centered at the origin. If $x = r \cos \theta$ and $y = r \sin \theta$, what is the value of $x^2 + y^2$?

The coordinates of the vertices of a triangle are $O(0,0)$, $A(a,0)$, and $B(0,b)$. If the distance from $O$ to the line $AB$ is $h$, which of the following represents $\frac{1}{h^2}$?

A cube has a surface area of $150$ cm$^2$. Calculate the length of the internal diagonal that connects two opposite vertices of the cube.

Two circles with radii $9$ cm and $4$ cm touch each other externally. A common external tangent touches the circles at points $P$ and $Q$. Find the length of the segment $PQ$.

A right-angled triangle has a perimeter of $40$ cm and a hypotenuse of length $17$ cm. Find the lengths of the two shorter sides.

Calculate the length of the hypotenuse $c$ in a right-angled triangle where the legs are $8$ and $15$.

In the triangle shown, find the angle $\theta$ to the nearest degree.

If $\tan \alpha = \frac{3}{4}$, find $\sin \alpha$ for an acute angle $\alpha$.

A surveyor stands 50 m from the base of a building. The angle of elevation to the top of the building is $35^{\circ}$. Find the height of the building.

In the figure, find the value of $y$.

A rhombus has diagonals of lengths $10$ cm and $24$ cm. Find the size of the smaller interior angle to the nearest degree.

In a right triangle $ABC$ with angle $C=90^{\circ}$, if $\cos A = 0.8$ and $AC = 16$, find the length of $BC$.

A flagpole of height $h$ stands on top of a building of height $B$. From a point $P$ on the ground, the angle of elevation to the bottom of the flagpole is $35^{\circ}$ and to the top of the flagpole is $45^{\circ}$. If the point $P$ is $20$ meters from the base of the building, find the height $h$ of the flagpole.

A right-angled triangle has a hypotenuse of length $13$ and a perimeter of $30$. Find the measure of the smallest angle in the triangle.

From the top of a tower of height $H$, the angle of depression to a point $A$ on the ground is $\alpha$. From a point halfway up the tower, the angle of depression to the same point $A$ is $\beta$. Find an expression for $\tan\alpha$ in terms of $\tan\beta$.

In a triangle, $a=3, b=5$ and $c=7$. Determine if the triangle is acute, right-angled, or obtuse.

The area of $\triangle PQR$ is $12$ cm$^2$. If $p=6$ and $q=8$, what is the sine of angle $R$?

Find the smallest angle of the triangle with sides 4, 5, and 6.

If $\sin A = 3/5$ and $A$ is obtuse, find $\cos A$.

In $\triangle ABC$, $\frac{a+b}{c} = \frac{\cos((A-B)/2)}{\sin(C/2)}$ is a known identity. If $A=90^\circ, B=60^\circ$ and $C=30^\circ$, verify the ratio $a:b:c$.

In $\triangle ABC$, angle $A = 60^\circ$ and $b:c = 2:3$. If the area is $6\sqrt{3}$, find the length of side $a$.

Find the length of the side $BC$ in $\triangle ABC$ if $AB=10$, $AC=10$ and angle $A=36^\circ$. (Hint: Golden triangle)

A vertical pole $OT$ of height $h$ stands at the center $O$ of a horizontal circular field. Three points $A, B, C$ are on the circumference of the field. The angles of elevation of the top of the pole $T$ from $A, B,$ and $C$ are all $\alpha$. If $AB = c$ and $\angle ACB = \gamma$, find an expression for $h$.

In a circle of radius $R = 12$ cm, a triangle $ABC$ is inscribed such that $\angle A = 50^{\circ}$ and $\angle B = 70^{\circ}$. Calculate the length of the side $BC$.

A ship $X$ is $15$ km from a lighthouse $L$ on a bearing of $310^{\circ}$. Another ship $Y$ is $22$ km from $L$ on a bearing of $040^{\circ}$. What is the distance between the two ships?

Determine the phase shift of the function $y = \sin(x - \pi/4)$ from the graph.

Which trigonometric identity is represented by the relationship between the sides of the triangle shown?

The graph of $y = \cos(x)$ is reflected in the x-axis. What is the equation of the resulting graph?

Find the value of $x$ in degrees for the first intersection of $y = \sin(x)$ and $y = 0.5$ for $x > 0$.

Identify the equation of the function shown, which has been translated horizontally and vertically.

For the curve $y = 3\sin(x) - 4\cos(x)$, what is the distance between the maximum and minimum y-values?

The identity $\tan x + \cot x$ is equivalent to which of the following?

The power $P$ in a circuit is given by $P = 2 \sin^2 \theta$. A student uses the identity $\cos(2\theta) = 1 - 2 \sin^2 \theta$ to transform the graph. Which of the following describes the graph of $P$ in terms of a standard cosine graph?

A sound wave is represented by $f(t) = \sin(t) - \sqrt{3} \cos(t)$. To analyze its interference, it is rewritten as $R \sin(t - \phi)$. Find the value of $\phi$ in radians where $0 < \phi < \pi$.

A pendulum's horizontal position $x$ is given by $x = 5 \sin(\pi t)$. A second pendulum's position is $y = 5 \cos(\pi t)$. Using the Pythagorean identity, determine the shape of the path traced by a point $(x, y)$ in the coordinate plane.

A point $B$ is 20 km from $A$ on a bearing of $150^\circ$. How far South is $B$ from $A$?

Given a triangle with sides 7 cm, 8 cm, and 9 cm, calculate the largest angle to the nearest degree.

A cube has a side length of $a$. What is the angle between a face diagonal and the base of the cube?

In the triangle shown, calculate the length of side $x$ using the Sine Rule.

A ship travels 30 km on a bearing of $040^\circ$ and then 40 km on a bearing of $130^\circ$. What is the direct distance from the starting point?

A vertical mast $PQ$ is supported by two wires $AP$ and $BP$ attached to the top $P$. $A$ and $B$ are on horizontal ground such that $QA = 10$ m, $QB = 15$ m, and angle $AQB = 80^\circ$. If the mast height is 20 m, find the distance $AB$.

The angle of elevation of a balloon from a point $A$ is $35^\circ$. From a point $B$, 1000 m closer to the balloon on the same horizontal level, the angle of elevation is $60^\circ$. Calculate the height of the balloon.

A flagpole $FG$ of height $10$ m stands vertically at the corner $F$ of a rectangular horizontal park $FGHJ$. The dimensions of the park are $FG = 30$ m and $GH = 40$ m. $P$ is a point on the diagonal $FH$ such that $FP:PH = 1:3$. Calculate the angle of elevation of the top of the flagpole $G$ from point $P$.

A right cone has a base radius of $5$ cm and a slant height of $13$ cm. $A$ and $B$ are two points on the circumference of the base such that the angle $AOB$ is $120^\circ$, where $O$ is the center of the base. Calculate the distance from $A$ to the vertex $V$ measured along the surface (the shortest path).

Two points $K$ and $L$ are on the same horizontal level as the foot of a hill. The bearing of the hilltop $H$ from $K$ is $045^\circ$ and the angle of elevation is $25^\circ$. The bearing of the hilltop $H$ from $L$ is $315^\circ$ and the angle of elevation is $20^\circ$. If $K$ and $L$ are $1000$ m apart, find the height of the hill.