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Fluidité · Pack A

9.3 Right-angle Trigonometry

Répondez à chaque question. Montrez les calculs si nécessaire.

Bronze

  1. In a right-angled triangle, the angle $\theta$ has opposite side 3 cm and hypotenuse 5 cm. Write down $\sin \theta$ as a fraction in its simplest form.

  2. Identify the opposite, adjacent, and hypotenuse for angle 30° in a right-angled triangle with hypotenuse 10 cm.

  3. Find sin 60° in surd form.

  4. A right triangle has opposite 5 and adjacent 12. Find tan θ.

  5. In a right triangle, $\sin \theta = 0.6$ and hypotenuse = 10. Find the opposite side.

  6. Use $\sin 30° = 0.5$ to find the opposite side when angle = 30° and hyp = 8 cm.

  7. Find $\tan 45°$ exactly.

  8. A right triangle has hypotenuse 10 and an angle 30°. Find the side opposite the 30° angle.

  9. Find tan 30° exactly.

  10. A right triangle has hyp 13 and one leg 12. Find the third side, and then $\sin \theta$ for the angle opposite the 5-side.

Silver

  1. In a right triangle, the side opposite $\theta$ is 5 cm and hyp = 12 cm. Find $\theta$ to 1 d.p.

  2. A right triangle has hyp 10 cm and angle 35°. Find the opposite side to 2 d.p.

  3. A right triangle has adjacent 8 cm and angle 30°. Find the opposite side to 2 d.p.

  4. A right triangle has adjacent 6 and opposite 8. Find (a) the hypotenuse, (b) the angle θ to 1 d.p.

  5. Find the angle of elevation: a tree casts a 12 m shadow when the tree is 7.5 m tall.

  6. A right triangle has hyp 10 and angle 30°. Find the adjacent (to 2 d.p.).

  7. A ladder of length 5 m leans against a wall, making a 60° angle with the ground. Find the height up the wall.

  8. A right triangle has opposite 8 and adjacent 15. Find (a) the angle θ to 1 d.p., (b) the hypotenuse.

  9. A 25 m vertical building. From the top, the angle of depression to a car is 20°. Find the horizontal distance.

  10. Find the angle in a right triangle with hyp 13 and opposite 5.

Gold

  1. A vertical tree casts a shadow of 12 m on horizontal ground. The angle of elevation from the tip of the shadow to the top of the tree is 32°. Find the tree height to 2 d.p.

  2. A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base.

  3. A right triangle has angle 30° and hyp 8 cm. Use exact values to find the opposite and adjacent.

  4. An aircraft's angle of approach to a runway is 3°. How high should it be when 7 km away horizontally? Give in m.

  5. A right triangle has hyp 10 and one acute angle 35°. (a) Find the opposite and adjacent. (b) Find the area.

  6. Find $x$ in: angle in a right triangle is $(x + 10)°$ and the opposite/adjacent ratio is 1.

  7. A right triangle has one acute angle of 25° and hypotenuse 20 cm. Find the perimeter to 2 d.p.

  8. Two right-angled triangles share a common leg. The first has opposite 3 and adjacent 4. The second has opposite 5 and the same adjacent 4. Find the difference in their hypotenuses.

  9. A right triangle has hyp 25 cm and an angle 60° at one acute vertex. Find the lengths of the two legs.

  10. A boat sails 10 km on bearing 030°. (a) How far north? (b) How far east?

Platinum

  1. PQRS is a square base 8 cm; pyramid apex V is 12 cm above the centre. M = midpoint of QR. Find (a) GM (G = centre), (b) VM, (c) angle of face VQR with the base.

  2. In a right triangle, the angle at A is 30° and the hypotenuse is 8 cm. Use exact values to find the legs.

  3. A cliff of height $h$ stands on level ground. From point A, 28 m from the base, line of sight to the top is 52 m. From B further, sight is 60 m. Find $h$ and the distance from B to the base.

  4. A flagpole is observed from two points 50 m apart. From the closer point, elevation = 35°. From the further, 22°. Find the height of the flagpole.

  5. A ladder of length 6 m leans against a wall. The angle with the ground is initially 70°; it slips to 60°. By how much does the top descend?

  6. Find $\sin 75°$ exactly using the identity $\sin(45° + 30°) = \sin 45 \cos 30 + \cos 45 \sin 30$.

  7. A triangular prism has a right-angled triangular cross-section with legs 6 and 8 cm. The hypotenuse is the base of the prism, lying on a table. Find the angle the prism's slanted face makes with the table.

  8. A square pyramid has slant edge from base vertex to apex = 13 cm and base diagonal half-length = 5 cm. Find the apex height above the centre.

  9. A skier descends a 100 m slope inclined at 15° to horizontal. Find (a) horizontal distance covered, (b) vertical drop.

  10. A 3D Pythagoras-with-trig problem: a square pyramid has base side 10 cm and slant edge 13 cm. Find the angle between a slant edge and the base.