Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the value of sin2θ + cos2θ.

The fundamental identity: sin2θ + cos2θ =

Perfect! This is the most important trigonometric identity.

(2) If sec θ = 1312, find tan θ.

Since sec θ = 1312, we have cos θ =

Using the required identity, we get tan θ =

Excellent! Remember to use the Pythagorean identity.

(3) Write the value of cot 60°.

cot 60° = 1332

Correct! cot 60° = 1tan60° = 13.

(4) Express tan θ in terms of sin θ and cos θ.

tan θ = sinθcosθcosθsinθsin θ × cos θsin θ + cos θ

Perfect! This is the fundamental definition of tangent.

(5) In ΔABC, right-angled at B, if AC = 13 cm and AB = 5 cm, find sin C.

First, find BC using Pythagoras: BC = cm

Therefore, sin C =

Excellent! sin C = oppositehypotenuse = ABAC.

(6) If tan θ = 3, what is the value of sin θ?

When tan θ = 3, the angle θ = °

Therefore, sin θ = 3212

Perfect! tan 60° = 3, so sin 60° = 32.

(7) If cot A = 34, find sin A and cos A.

Since cot A = 34, we have tan A =

Using a right triangle with opposite = and adjacent = , hypotenuse =

Therefore: sin A = and cos A =

Excellent! Always draw a right triangle to visualize the ratios.

Short Answer Questions (2 Marks Each)

(1) If cos θ = 45 and θ is acute, find sin θ and tan θ.

Since θ is acute, sin θ =

Therefore, tan θ =

Perfect! Always check if the angle is acute or obtuse for the sign.

(2) Find the height of a tree if the length of its shadow is 5 m and the angle of elevation of the Sun is 60°.

Let height = h. Then tan 60° =

Since tan 60° = 3, we get h = 5353 meters = meter (Upto two decimal places)

Excellent! Height = 53 ≈ 8.66 meters.

(3) From a point on the ground, the angle of elevation of the top of a tower is 30°. The height of the tower is 30 m. Find the distance of the point from the foot of the tower.

Let distance = d. Then tan 30° =

Since tan 30° = 13, we get d = 303303 meters = meter (Upto two decimal places)

Perfect! Distance = 303 ≈ 51.96 meters.

(4) If tan A = 34, find the values of the other trigonometric ratios.

Using a right triangle: opposite = , adjacent = , hypotenuse =

sin A = , cos A = , cot A =

sec A = , csc A =

Excellent! All six trigonometric ratios found systematically.

Long Answer Questions (4 Marks Each)

(1) A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a lighthouse as 60° and the angle of depression of its base as 30°. Find the height of the lighthouse.

Total height of lighthouse = meters

Excellent! The lighthouse is 40 meters tall.

(2) A boy is flying a kite with a string of length 100 m, making an angle of 45° with the ground. Find the height of the kite, assuming there is no slack.

Height of Kite = 5021002 meters = meters (Upto two decimal places)

Perfect! Height = 502 ≈ 70.71 meters.

(3) Two poles of heights 12 m and 20 m are 30 m apart. Find the distance between their tops.

Distance between tops = 22412175 meters ≈ meters (Upto two decimal places)

Excellent problem-solving! The distance is 2√241 meters.

Part B: Objective Questions (1 Mark Each)

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

(1) If tan A = 1, then angle A is:

(a) 30° (b) 60° (c) 45° (d) 90°

Correct! tan 45° = 1.

(2) If cos θ = 0.6, then sin θ =

(a) 0.4 (b) 0.8 (c) 1.2 (d) 1.6

Correct! Using sin2θ + cos2θ = 1, sin θ = 0.8.

(3) Which identity is true?

(a) tan2θ – sec2θ = 1 (b) sin2θ + cos2θ = 1 (c) cot2θ + cosec2θ = 1 (d) tan2θ + cot2θ = 1

Correct! This is the fundamental trigonometric identity.

(4) If sin θ = 513, then cos θ =

(a) 1213 (b) 512 (c) 135 (d) 1

Correct! Using the 5-12-13 Pythagorean triple.

(5) If the height and shadow of a building are equal, the angle of elevation is:

(a) 30° (b) 60° (c) 45° (d) 90°

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

(6) If tan θ = 13, then θ =

(a) 30° (b) 45° (c) 60° (d) 90°

Correct! tan 30° = 13.

(7) The value of sec 30° is:

(a) 1 (b) 23 (c) 32 (d) 2

Correct! sec 30° = 1cos30° = 132 = 23.

(8) The trigonometric ratio which is always greater than 1 for acute angle is:

(a) sin θ (b) cos θ (c) tan θ (d) sec θ

Correct! sec θ = 1cosθ, and since cos θ < 1 for acute angles, sec θ > 1.

(9) What is the value of sin230° + cos230°?

(a) 0.75 (b) 1 (c) 1.5 (d) 2

Correct! For any angle, sin2θ+cos2θ = 1.

(10) If tan θ = sinθcosθ, then cot θ =

(a) sinθcosθ (b) cosθsinθ (c) 1tanθ (d) Both B and C

Correct! cot θ = cosθsinθ = 1tanθ.

Trigonometry Challenge

Determine whether these statements about trigonometry are True or False:

Trigonometry Quiz