Trigonometry Foundations Worksheet

This worksheet covers the Trigonometry Foundations lessons: Angles and Triangles, Pythagoras' Theorem, The Three Ratios, Finding Missing Sides, Finding Missing Angles, and Angles of Elevation and Depression.

There are $10$ questions in each of the $6$ sections ($60$ questions in total). Show your working where a calculation is involved. Work on paper, then check against the lessons if you get stuck.

Q1. A triangle has angles of $50^{\circ }$ and $60^{\circ }$. Find the third angle.

Q2. A triangle has angles of $90^{\circ }$ and $35^{\circ }$. Find the third angle.

Q3. Two angles on a straight line are $115^{\circ }$ and $x$. Find $x$.

Q4. Three angles around a point are $120^{\circ }$, $95^{\circ }$, and $y$. Find $y$.

Q5. An isosceles triangle has a base angle of $70^{\circ }$. Find the other base angle and the apex angle.

Q6. A triangle has angles of $42^{\circ }$, $42^{\circ }$, and $z$. Find $z$.

Q7. A right-angled triangle has one other angle of $28^{\circ }$. Find the third angle.

Q8. An equilateral triangle has three equal angles. Find each angle.

Q9. Two equal angles lie on a straight line. Find each angle.

Q10. A triangle has angles in the ratio $2:3:4$. Find all three angles.

Q1. A right-angled triangle has legs of $3$ cm and $4$ cm. Find the length of the hypotenuse.

Q2. A right-angled triangle has legs of $6$ cm and $8$ cm. Find the length of the hypotenuse.

Q3. A right-angled triangle has legs of $5$ cm and $12$ cm. Find the length of the hypotenuse.

Q4. A right-angled triangle has legs of $9$ cm and $12$ cm. Find the length of the hypotenuse.

Q5. A right-angled triangle has a hypotenuse of $13$ cm and one leg of $5$ cm. Find the other leg.

Q6. A right-angled triangle has a hypotenuse of $10$ cm and one leg of $6$ cm. Find the other leg.

Q7. A right-angled triangle has legs of $7$ cm and $24$ cm. Find the length of the hypotenuse.

Q8. A right-angled triangle has a hypotenuse of $17$ cm and one leg of $8$ cm. Find the other leg.

Q9. A right-angled triangle has legs of $1.5$ cm and $2$ cm. Find the length of the hypotenuse.

Q10. A right-angled triangle has a hypotenuse of $25$ cm and one leg of $7$ cm. Find the other leg.

Q1. In a right-angled triangle, the side opposite angle $\theta$ is $3$ cm and the hypotenuse is $5$ cm. Find $\sin (\theta )$.

Q2. In a right-angled triangle, the side adjacent to angle $\theta$ is $4$ cm and the hypotenuse is $5$ cm. Find $\cos (\theta )$.

Q3. In a right-angled triangle, the side opposite angle $\theta$ is $3$ cm and the side adjacent is $4$ cm. Find $\tan (\theta )$.

Q4. In a right-angled triangle, the side opposite angle $\theta$ is $5$ cm and the hypotenuse is $13$ cm. Find $\sin (\theta )$.

Q5. In a right-angled triangle, the side adjacent to angle $\theta$ is $12$ cm and the hypotenuse is $13$ cm. Find $\cos (\theta )$.

Q6. In a right-angled triangle, the side opposite angle $\theta$ is $5$ cm and the side adjacent is $12$ cm. Find $\tan (\theta )$.

Q7. In a right-angled triangle, the side opposite angle $\theta$ is $8$ cm and the hypotenuse is $17$ cm. Find $\sin (\theta )$.

Q8. In a right-angled triangle, the side adjacent to angle $\theta$ is $15$ cm and the hypotenuse is $17$ cm. Find $\cos (\theta )$.

Q9. In a right-angled triangle, the side opposite angle $\theta$ is $8$ cm and the side adjacent is $15$ cm. Find $\tan (\theta )$.

Q10. In a right-angled triangle, the side opposite angle $\theta$ is $7$ cm and the hypotenuse is $25$ cm. Find $\sin (\theta )$.

Q1. A right-angled triangle has a hypotenuse of $10$ cm and one angle of $30^{\circ }$. Find the length of the side opposite this angle.

Q2. A right-angled triangle has a hypotenuse of $14$ cm and one angle of $60^{\circ }$. Find the length of the side adjacent to this angle.

Q3. A right-angled triangle has a side of $9$ cm adjacent to a $45^{\circ }$ angle. Find the length of the side opposite this angle.

Q4. A right-angled triangle has a side of $6$ cm opposite a $30^{\circ }$ angle. Find the length of the hypotenuse.

Q5. A right-angled triangle has a side of $8$ cm adjacent to a $60^{\circ }$ angle. Find the length of the hypotenuse.

Q6. A right-angled triangle has a hypotenuse of $20$ cm and one angle of $30^{\circ }$. Find the length of the side opposite this angle.

Q7. A right-angled triangle has a hypotenuse of $16$ cm and one angle of $60^{\circ }$. Find the length of the side adjacent to this angle.

Q8. A right-angled triangle has a side of $15$ cm opposite a $45^{\circ }$ angle. Find the length of the side adjacent to this angle.

Q9. A right-angled triangle has a hypotenuse of $24$ cm and one angle of $30^{\circ }$. Find the length of the side adjacent to this angle.

Q10. A right-angled triangle has a side of $12$ cm adjacent to a $45^{\circ }$ angle. Find the length of the hypotenuse.

Q1. In a right-angled triangle, the side opposite angle $\theta$ is $3$ cm and the hypotenuse is $5$ cm. Find the angle $\theta$, using $\sin$.

Q2. In a right-angled triangle, the side adjacent to angle $\theta$ is $4$ cm and the hypotenuse is $5$ cm. Find the angle $\theta$, using $\cos$.

Q3. In a right-angled triangle, the side opposite angle $\theta$ is $5$ cm and the side adjacent is $12$ cm. Find the angle $\theta$, using $\tan$.

Q4. In a right-angled triangle, the side opposite angle $\theta$ is $8$ cm and the hypotenuse is $17$ cm. Find the angle $\theta$, using $\sin$.

Q5. In a right-angled triangle, the side opposite angle $\theta$ is $7$ cm and the side adjacent is $24$ cm. Find the angle $\theta$, using $\tan$.

Q6. In a right-angled triangle, the side opposite angle $\theta$ is $20$ cm and the hypotenuse is $29$ cm. Find the angle $\theta$, using $\sin$.

Q7. In a right-angled triangle, the side adjacent to angle $\theta$ is $21$ cm and the hypotenuse is $29$ cm. Find the angle $\theta$, using $\cos$.

Q8. In a right-angled triangle, the side opposite angle $\theta$ is $1$ cm and the hypotenuse is $1.41$ cm. Find the angle $\theta$, using $\sin$.

Q9. In a right-angled triangle, the two legs are both $1$ cm. Find the angle $\theta$ between the hypotenuse and one leg, using $\tan$.

Q10. In a right-angled triangle, the side adjacent to angle $\theta$ is $1$ cm and the hypotenuse is $2$ cm. Find the angle $\theta$, using $\cos$.

Q1. A person stands $20$ $m$ from the base of a tower. The angle of elevation to the top of the tower is $30^{\circ }$. Find the height of the tower.

Q2. A person stands $15$ $m$ from a building. The angle of elevation to the top of the building is $45^{\circ }$. Find the height of the building.

Q3. A person stands $50$ $m$ from a flagpole. The angle of elevation to the top of the flagpole is $20^{\circ }$. Find the height of the flagpole.

Q4. From the top of a cliff $40$ $m$ high, the angle of depression to a boat is $25^{\circ }$. Find the horizontal distance from the base of the cliff to the boat.

Q5. A ladder leans against a wall, making an angle of $60^{\circ }$ with the ground. The foot of the ladder is $3$ $m$ from the wall. Find how high up the wall the ladder reaches.

Q6. A person $1.5$ $m$ tall stands $30$ $m$ from a tower and measures the angle of elevation to the top of the tower as $35^{\circ }$. Find the total height of the tower.

Q7. From a lighthouse $60$ $m$ tall, the angle of depression to a ship is $15^{\circ }$. Find the distance from the base of the lighthouse to the ship.

Q8. A person stands $25$ $m$ from a tree. The angle of elevation to the top of the tree is $40^{\circ }$. Find the height of the tree.

Q9. An observer on a cliff $25$ $m$ high sees a boat at an angle of depression of $18^{\circ }$. Find the distance from the base of the cliff to the boat.

Q10. A person $1.6$ $m$ tall stands $12$ $m$ from a pole and measures the angle of elevation to the top of the pole as $50^{\circ }$. Find the total height of the pole.