Trigonometry

In the triangle below, find the value of $y$ using $\cos(45^{\circ}) \approx 0.707$.

What is $\tan(\theta)$ for the angle shown?

Find the length of side $a$ in the triangle below, given $\sin(30^{\circ}) = 0.5$.

In triangle $PQR$, find $\sin(Q)$.

A boat travels $10$ km on a bearing of $060^{\circ}$. How far north has it traveled from its starting point? (Recall $\cos(30^{\circ}) = \sin(60^{\circ}) \approx 0.866$ and $\sin(30^{\circ}) = \cos(60^{\circ}) = 0.5$)

In a right-angled triangle, the side opposite to angle $\phi$ is $12$ cm and the adjacent side is $5$ cm. Find $\sin(\phi)$.

Find the length $d$ in the triangle. (Use $\cos(55^{\circ}) \approx 0.57$)

A tree casts a shadow $22$ m long when the sun's angle of elevation is $28^{\circ}$. How much shorter will the shadow be when the sun's angle of elevation increases to $42^{\circ}$? Round to the nearest meter.

A rectangle has a length of $15$ cm and a diagonal of $17$ cm. Calculate the angle between the diagonal and the $15$ cm side to the nearest tenth of a degree.

A point $S$ is $15$ km North and $20$ km West of a lighthouse $L$. Calculate the bearing of $S$ from $L$ to the nearest degree.

Identify which equation is correct for the given triangle.

Calculate the length of $DE$.

In the triangle below, find the value of $k$.

Determine the size of angle $x$ in the diagram. (Round to 1 decimal place)

A ramp has a length of $6$ m and rises to a height of $1.5$ m. What is the angle of inclination of the ramp?

Calculate the perimeter of the right-angled triangle $ABC$ if $\angle B = 30^\circ$ and $BC = 10$ cm.

Find the length of $AC$ in the rectangular box (cuboid) shown below. (Round to 1 d.p.)

A rhombus has side lengths of $8$ cm and one of its internal angles is $120^{\circ}$. Calculate the length of the shorter diagonal.

A tower is supported by two wires, $A$ and $B$, attached to the same point at the top of the tower. Wire $A$ is $20$ m long and makes an angle of $60^{\circ}$ with the ground. Wire $B$ is attached to the ground $10$ m further away from the tower than wire $A$. Calculate the angle wire $B$ makes with the ground. (Assume both wires are on the same side of the tower).

A wheelchair ramp is to be built with a constant slope. The ramp must rise $0.8$ m over a horizontal distance of $5$ m. However, safety regulations state the angle of the ramp must not exceed $8^{\circ}$. Calculate the actual angle of the ramp and determine if it meets the safety standard.