Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) What is the trigonometric ratio used when you have the height and hypotenuse of a right-angled triangle?

Perfect! Sine relates the opposite side (height) to the hypotenuse.

(2) State whether the angle of elevation increases or decreases when the observer moves closer to the object.

Correct! Moving closer makes the object appear higher, increasing the angle.

(3) Give an example of an object that can be measured using the angle of depression. Object belowObject Above

Excellent! Angle of depression is used for objects below the horizontal line.

(4) What is the value of tan 45°? tan 45° =

Perfect! This is one of the standard angle values.

(5) What does an angle of elevation measure in a practical scenario? The angle between the line and the line of sight to an object the observer

Excellent understanding of angle of elevation!

Short Answer Questions (2 Marks Each)

(1) From a point 36 m away from the foot of a tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower. h = 363203 m

Perfect! Standard application of tangent ratio.

(2) The height of a lighthouse is 30 m. The angle of depression of a boat from the top of the lighthouse is 45°. Find the distance of the boat from the base of the lighthouse. d = m

Excellent! Angle of depression equals angle of elevation from the boat.

(3) A man 1.7 m tall observes the top of a building at an angle of elevation of 30°. The building is 37.7 m tall. Find the distance between the man and the building. d = 363363 m

Perfect! Remembered to subtract the observer's height.

(4) From a certain point on the ground, the angles of elevation of the top and bottom of a water tank placed on a tower are 60° and 45° respectively. If the height of the tower is 15 m, find the height of the water tank. h = 153−1153+1 m

Excellent! Used both angles to find the tank height.

(5) Two buildings of different heights are 50 m apart. The angle of elevation from the top of the shorter building to the top of the taller one is 30°. If the shorter building is 20 m high, find the height of the taller building. Height of taller building = + 5033503 m

Perfect! Found the height difference first, then added to shorter building height.

(6) The angle of depression of the top and bottom of an apartment building from the top of a hill 72 m high are 30° and 60° respectively. Find the height of the building. h = m

Excellent! The building height is 48 m.

Long Answer Questions (4 Marks Each)

(1) The angles of elevation of the top of a tower from two points at distances 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.

(2) The angle of elevation of the top of a tower from a point on the ground is 30°. On walking 80 m towards the tower, the angle of elevation becomes 60°. Find the height of the tower and the distance of the original point from the base. d = m and h = 403203 m

Excellent! Tower height = 403 m, original distance = 120 m.

(3) From the top of a building 60 m high, the angles of depression of the top and bottom of a lamp post are observed to be 30° and 60° respectively. Find the height of the lamp post. h = m

Perfect! The lamp post height is 40 m.

(4) A man on top of a tower finds the angle of depression of a point on the ground to be 45°. He walks down 30 m and finds that the angle of depression becomes 60°. Find the height of the tower. H = 153+1153−1 m

Excellent! Tower height = 153+1 m ≈ 40.98 m.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) If a tower casts a shadow of length 53 m when the angle of elevation is 30°, then the height of the tower is:

(a) 5 m (b) 10 m (c) 15 m (d) 20 m

Correct! Using tan 30° = h53: 13 = h53, so h = 5 m.

(2) A building of height 103 m casts a shadow of length 10 m. The angle of elevation of the sun is:

(a) 30° (b) 45° (c) 60° (d) 90°

Correct! tan θ = 10310 = 3, so θ = 60°.

(3) In a right triangle, if tan A = 1, then angle A is:

(a) 30° (b) 60° (c) 45° (d) 90°

Correct! tan 45° = 1 is a standard angle value.

(4) The angle of depression is equal to:

(a) Angle of elevation (b) Angle formed with vertical (c) Angle formed with horizontal line (d) Angle between object and eye

Correct! Angle of depression is measured below the horizontal line.

(5) If a person observes an object at an angle of elevation of 60° and is 20 m away from the object, the object's height is:

(a) 103 (b) 203 (c) 40 (d) 10

Correct! Using tan 60° = h20: 3 = h20, so h = 203 m.

(6) A 50 m tall tower casts a 50 m long shadow. The angle of elevation of the sun is:

(a) 30° (b) 45° (c) 60° (d) 90°

Correct! tan θ = 5050 = 1, so θ = 45°.

(7) The height of a tree is h and the shadow is also h. What is the angle of elevation of the sun?

(a) 45° (b) 60° (c) 30° (d) 90°

Correct! When height equals shadow length, tan θ = hh = 1, so θ = 45°.

(8) If tan θ = 13, then θ =

(a) 30° (b) 45° (c) 60° (d) 90°

Correct! tan 30° = 13 is a standard angle value.

(9) If the angle of elevation is 60° and base is 6 m, then height is:

(a) 63 m (b) 63 m (c) 33 m (d) 3 m

Correct! Using tan 60° = h6: 3 = h6, so h = 63 m.

(10) The observer sees the top of a tower at 30° and moves 50 m closer to it, the angle becomes 60°. The height of the tower is:

(a) 25 m (b) 43.3 m (c) 50 m (d) 86.6 m

Correct! Using the standard two-angle problem: h = 5033−1 ≈ 43.3 m.

Trigonometry Applications Challenge

Determine whether these statements about trigonometry applications are True or False:

Trigonometry Applications Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Angle Concepts: Understanding angles of elevation and depression in real-world contexts

(2) Height-Distance Problems: Using trigonometric ratios to find unknown measurements

(3) Standard Angle Applications: Applying values of 30°, 45°, 60° in practical problems

(4) Complex Scenarios: Solving problems with multiple observation points and angles

(5) Shadow Problems: Relating sun's angle to shadow lengths and object heights

(6) Complementary Angles: Using the relationship between angles that sum to 90°

(7) Problem-solving Strategies: Systematic approaches to multi-step trigonometric problems

(8) Real-world Modeling: Translating practical situations into mathematical equations

Excellent work mastering the practical applications of trigonometry in real-world scenarios!