Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) If sec θ = 52, find tan θ. tan θ =

Perfect! Used the fundamental trigonometric identity: sec2θ - tan2θ = 1: tan2θ = sec2θ - 1 = 522 - 1 = 14, so tan θ = 12

(2) Write the value of cos 30° – sin 60°.

Excellent! These special angles have equal values.

(3) If sin A = cos B, then A + B = ? °

Perfect! Used the complementary angle relationship.

(4) Evaluate: 1+tan2Asec2A

Excellent application of the fundamental identity! Using identity 1+tan2A = sec2A: 1+tan2Asec2A = sec2Asec2A = 1

(5) Write the value of cot60°+tan30°2.

Perfect calculation with special angles!

(6) If sin θ = 0.6 and θ is acute, find cos θ without using a calculator. cos θ =

Excellent! For acute angles, cosine is positive.

(7) Express sec2A – tan2A as an identity. sec2A - tan2A =

Perfect! This is one of the fundamental trigonometric identities.

Short Answer Questions (2 Marks Each)

Note: Answer each question with steps and explanation, in 2-3 sentences. Write down the answers on sheet and submit to the school subject teacher.

(1) Prove: sin A + cos Asin A - cos A + sin A - cos Asin A + cos A = 2sin2A+cos2Asin2A−cos2A

(2) From a point 120 m away from the foot of a tower, the angle of elevation of the top is 30°. Find the height of the tower. Tower Height = m

Perfect application of trigonometry in height-distance problems!

(3) If cot A = 724, show that 1+sinA1−sinA = cos2A.

LHS =

RHS =

Excellent verification using trigonometric relationships!

(4) A man observes the top of a building at an angle of elevation of 45°. He moves 30 m towards the building and the angle becomes 60°. Find the height of the building. Building Height = + m

Excellent problem involving two angles of elevation!

(5) If sec θ + tan θ = p, prove that sec θ = p2+12p.

(6) Show that: 1−tanA1+tanA = cosA−sinAcosA+sinA.

Long Answer Questions (4 Marks Each)

Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.

(1) The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 20 m nearer to the tower, the angle of elevation is 45°. Find the height of the tower and the distance of the first point from the tower.

Height of tower = m

Distance of first point = m (upto two decimal places)

Excellent solution involving two angles of elevation!

(2) Prove the identity: 1 + cos Asin A + 1 - cos Asin A = 2 cosec A.

(3) A balloon is flying at a height of 120 m. The angle of elevation from two points on the ground (on the same side of the balloon) are 30° and 60°. Find the distance between the two points.

Distance between points = m (upto two decimal places)

Excellent application with two observation points!

(4) From the top of a building 50 m high, the angle of depression of the top and bottom of a tower are observed to be 30° and 60° respectively. Find the height of the tower.

Height of tower = m (upto two decimal places)

Excellent problem involving angles of depression!

Part B: Objective Questions (1 Mark Each)

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

(1) If sin θ = 35, then cos2θ =

(a) 925 (b) 1625 (c) 1225 (d) 45

9/25

16/25

12/25

4/5

Correct! Using sin2θ + cos2θ = 1: cos2θ = 1 - 352 = 1 - 925 = 1625.

(2) Which of the following is equal to 1?

(a) tan2θ + cot2θ (b) sin2θ – cos2θ (c) sec2θ – tan2θ (d) sin θ + cos θ

tan²θ + cot²θ

sin²θ – cos²θ

sec²θ – tan²θ

sin θ + cos θ

Correct! This is the fundamental identity sec2θ−tan2θ = 1.

(3) The value of cot230°−tan260° is:

(a) 1 (b) -1 (c) 0 (d) 2

-1

Correct!

(4) If sec2A=1+tan2A, then tan2A =

(a) sec2A−1 (b) sec2A+1 (c) 1−sec2A (d) 2sec2A

sec²A – 1

sec²A + 1

1 – sec²A

2sec²A

Correct! Rearranging the identity: tan2A = sec2A−1.

(5) If sin θ = 513, then tan θ =

(a) 125 (b) 512 (c) 135 (d) 1213

12/5

5/12

13/5

12/13

Correct! cos θ = 1213 (using Pythagorean identity), so tan θ = sinθcosθ = 5131213 = 512.

(6) Value of tan245° – cot245° is:

(a) 0 (b) 1 (c) 2 (d) –1

–1

Correct! tan 45° = 1 and cot 45° = 1, so 12 - 12 = 0.

(7) If cos θ = 0.8, then sin2θ =

(a) 0.36 (b) 0.64 (c) 0.96 (d) 0.8

0.36

0.64

0.96

0.8

Correct! Using sin2θ + cos2θ = 1: sin2θ = 1 - 0.82 = 1 - 0.64 = 0.36.

(8) Which is correct for all θ?

(a) sin2θ−cos2θ = 1 (b) sin2θ+cos2θ = 1 (c) tan2θ−sec2θ = 1 (d) sec2θ+tan2θ = 1

sin²θ – cos²θ = 1

sin²θ + cos²θ = 1

tan²θ – sec²θ = 1

sec²θ + tan²θ = 1

Correct! This is the most fundamental trigonometric identity.

(9) tan A = 1 implies angle A is:

(a) 30° (b) 60° (c) 90° (d) 45°

30°

60°

90°

45°

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

(10) Which is equal to cot A?

(a) sinAcosA (b) cosAsinA (c) 1secA (d) 1tanA

sin A/cos A

cos A/sin A

1/sec A

1/tan A

Correct! cot A = cosAsinA. Note that option (d) is also correct since cot A = 1tanA.

sin²θ + cos²θ = 1

Height/Distance problems

tan 30° = 1/√3

sin 60° = √3/2

sec²θ - tan²θ = 1

Tower height calculations

Angle of elevation/depression

1 + tan²θ = sec²θ

cos 45° = 1/√2

Trigonometric Identities

Standard Angle Values

Applications

Trigonometry Challenge

Determine whether these statements about trigonometry are True or False:

sin²θ + cos²θ = 1 for all θ

sec²θ - tan²θ = 1

sin 30° = √3/2

If sin A = cos B, then A + B = 90°

cot θ = tan θ

tan θ = sin θ × cos θ

Trigonometry Quiz

🎉 You Did It! What You've Learned:

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

(1) Fundamental Identities: sin2θ+cos2θ = 1, sec2θ−tan2θ = 1, and their applications

(2) Standard Angle Values: Exact values for 30°, 45°, 60°, and 90° angles

(3) Trigonometric Relationships: Converting between different trigonometric functions

(4) Height and Distance Problems: Using angles of elevation and depression

(5) Identity Proofs: Algebraic manipulation to prove trigonometric identities

(6) Real-world Applications: Tower heights, building problems, and distance calculations

(7) Problem-solving Strategies: Systematic approach to complex trigonometric problems

(8) Advanced Techniques: Working with multiple angles and composite problems

Excellent work mastering advanced trigonometry concepts and their practical applications!

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