SSC Trigonometry Questions with Solutions and Formulas

Trigonometry is a high-weightage topic in SSC CGL, CHSL and CPO exams. In Tier 1, two to three questions come from it, and in Tier 2 the number is higher. The questions are based on a few standard values, about ten identities and the heights and distances setup. This practice paper has 30 trigonometry questions with answers and short solutions to help you master the topic quickly.

Practice Paper at a Glance

What Is Trigonometry? Basic Concept in Simple Words

Trigonometry studies the relationship between the sides and angles of a right-angled triangle. For an acute angle theta in a right triangle:

• sin theta = Perpendicular / Hypotenuse
• cos theta = Base / Hypotenuse
• tan theta = Perpendicular / Base
• cosec theta = 1 / sin theta, sec theta = 1 / cos theta and cot theta = 1 / tan theta

Trigonometry Values, Identities and Shortcut Tricks for SSC

• Identities: sin^2 + cos^2 = 1, 1 + tan^2 = sec^2 and 1 + cot^2 = cosec^2.
• Complementary angles: sin(90 - theta) = cos theta, tan(90 - theta) = cot theta and sec(90 - theta) = cosec theta.
• If A + B = 90, then tan A x tan B = 1 and sin^2 A + sin^2 B = 1.
• Memory trick for sin: sin values for 0, 30, 45, 60 and 90 are the square roots of 0, 1, 2, 3, 4 divided by 2. Cos values are the same in reverse.
• tan theta + cot theta = 1 / (sin theta x cos theta).
• If a + 1/a = 2 style questions: if tan theta + cot theta = 2, then theta = 45 degrees.
• Heights and distances: angle of elevation (looking up) and angle of depression (looking down) are measured from the horizontal. Use tan = height / distance in most questions.
• Useful pair: (sec theta + tan theta)(sec theta - tan theta) = 1.

Set 1: Standard Values and Basic Ratios (Q1–Q10)

Q1. Find the value of sin 30 degrees + cos 60 degrees.
Answer: 1
Solution: 1/2 + 1/2 = 1.

Q2. Find the value of tan 45 degrees + sin 90 degrees - cos 0 degrees.
Answer: 1
Solution: 1 + 1 - 1 = 1.

Q3. Find the value of sin 60 degrees x cos 30 degrees + cos 60 degrees x sin 30 degrees.
Answer: 1
Trick: (root 3 / 2)(root 3 / 2) + (1/2)(1/2) = 3/4 + 1/4 = 1.

Q4. Find the value of sin 30 degrees x cos 60 degrees.
Answer: 1/4
Solution: (1/2) x (1/2) = 1/4.

Q5. Find the value of tan 60 degrees x tan 30 degrees.
Answer: 1
Solution: root 3 x (1 / root 3) = 1.

Q6. If sin theta = 3/5, find cos theta (theta is acute).
Answer: 4/5
Solution: cos theta = square root of (1 - 9/25) = square root of (16/25) = 4/5.

Q7. If tan theta = 3/4, find sin theta.
Answer: 3/5
Trick: Perpendicular = 3 and base = 4, so hypotenuse = 5. sin theta = 3/5.

Q8. Find the value of sec^2 35 degrees - tan^2 35 degrees.
Answer: 1
Solution: From the identity 1 + tan^2 = sec^2, the difference is 1.

Q9. Find the value of cosec 30 degrees.
Answer: 2
Solution: sin 30 = 1/2, so cosec 30 = 2.

Q10. Find the value of cot 45 degrees + cosec 30 degrees.
Answer: 3
Solution: cot 45 = 1 and cosec 30 = 2, so the sum is 3.

Set 2: Identities and Complementary Angles (Q11–Q20)

Q11. If sin theta = cos theta for an acute angle, find theta.
Answer: 45 degrees
Solution: sin theta = cos theta means tan theta = 1, so theta = 45 degrees.

Q12. Find the value of sin 25 degrees / cos 65 degrees.
Answer: 1
Trick: cos 65 = cos(90 - 25) = sin 25. So the ratio = 1.

Q13. Find the value of tan 15 degrees x tan 75 degrees.
Answer: 1
Trick: 75 = 90 - 15, so tan 75 = cot 15. The product = 1.

Q14. Find the value of cos 1 x cos 2 x cos 3 x ... x cos 90 (all in degrees).
Answer: 0
Trick: The product contains cos 90 degrees = 0, so the product is 0.

Q15. Find the value of (sin theta + cos theta)^2 + (sin theta - cos theta)^2.
Answer: 2
Trick: Expanding, 2 sin^2 theta + 2 cos^2 theta = 2.

Q16. If tan theta + cot theta = 2, find tan^2 theta + cot^2 theta.
Answer: 2
Solution: Squaring: tan^2 + cot^2 + 2 = 4, so tan^2 + cot^2 = 2.

Q17. Find the value of (1 + tan^2 theta) x cos^2 theta.
Answer: 1
Solution: 1 + tan^2 = sec^2, and sec^2 x cos^2 = 1.

Q18. If cos theta = 5/13, find tan theta.
Answer: 12/5
Solution: sin theta = 12/13. tan theta = sin / cos = (12/13) / (5/13) = 12/5.

Q19. If sin A = 8/17, find sec A.
Answer: 17/15
Solution: cos A = 15/17, so sec A = 17/15.

Q20. If sec theta + tan theta = 2, find sec theta - tan theta.
Answer: 1/2
Trick: (sec + tan)(sec - tan) = 1, so sec - tan = 1/2.

Set 3: Heights and Distances and Mixed Questions (Q21–Q30)

Q21. A 10 m long ladder makes an angle of 30 degrees with the ground. How high does it reach on the wall?
Answer: 5 m
Solution: Height = 10 x sin 30 = 10 x 1/2 = 5 m.

Q22. A pole 10 m high casts a shadow of 10 root 3 m. Find the angle of elevation of the sun.
Answer: 30 degrees
Solution: tan theta = 10 / (10 root 3) = 1 / root 3, so theta = 30 degrees.

Q23. From a point 30 m away from the foot of a tower, the angle of elevation of its top is 45 degrees. Find the height of the tower.
Answer: 30 m
Solution: Height = 30 x tan 45 = 30 m.

Q24. From a point 20 m from the foot of a tower, the angle of elevation of the top is 60 degrees. Find the height of the tower.
Answer: 20 root 3 m (about 34.6 m)
Solution: Height = 20 x tan 60 = 20 root 3 m.

Q25. The angle of elevation of the top of a tower from a point 50 m away from its foot is 30 degrees. Find the height.
Answer: 50 / root 3 m (about 28.9 m)
Solution: Height = 50 x tan 30 = 50 / root 3 m.

Q26. From the top of a 60 m high tower, the angle of depression of a ship is 30 degrees. How far is the ship from the foot of the tower?
Answer: 60 root 3 m (about 103.9 m)
Solution: Distance = 60 / tan 30 = 60 root 3 m.

Q27. A kite string of length 100 m is inclined at 30 degrees to the horizontal. Find the height of the kite (assume the string is straight).
Answer: 50 m
Solution: Height = 100 x sin 30 = 50 m.

Q28. If A + B = 90 degrees, find the value of tan A x tan B + sin^2 A + sin^2 B.
Answer: 2
Trick: tan A x tan B = 1 and sin^2 A + sin^2 B = sin^2 A + cos^2 A = 1. Total = 2.

Q29. Find the value of sin^2 30 degrees + cos^2 60 degrees + tan^2 45 degrees.
Answer: 1.5
Solution: 1/4 + 1/4 + 1 = 1.5.

Q30. In triangle ABC, the right angle is at B, AB = 5 cm and BC = 12 cm. Find sin A.
Answer: 12/13
Solution: AC = 13 cm. sin A = BC / AC = 12/13.

Common Mistakes in Trigonometry

• Mixing up the values of sin 30 and cos 30 or sin 60 and cos 60.
• Forgetting that tan 90 and cot 0 are not defined.
• Using the wrong identity (for example sec^2 = 1 - tan^2).
• Taking the angle of depression from the vertical instead of the horizontal.
• Not checking whether the angle is acute before taking a positive square root.
• Writing tan = base / perpendicular by mistake.

How to Practice These 30 Questions

1. Memorise the table of standard values using the square-root trick.
2. Revise the identities and complementary angle pairs.
3. Solve Set 1 and Set 2 in 20 minutes and Set 3 in 12 minutes.
4. Draw a right triangle for every heights and distances question.
5. Attempt previous year SSC trigonometry questions after this paper.

Related Topics to Study Next

• Geometry Questions
• Mensuration Questions
• SSC Algebra Questions
• SSC Previous Year Papers

Frequently Asked Questions

How many trigonometry questions come in SSC CGL?

Two to three questions come from trigonometry in Tier 1 and more in Tier 2. Heights and distances, identities and standard values are the most common types.

What are the basic trigonometric ratios?

sin = perpendicular / hypotenuse, cos = base / hypotenuse and tan = perpendicular / base. The reciprocals are cosec, sec and cot respectively.

What are the standard values of sin, cos and tan?

sin 0, 30, 45, 60, 90 = 0, 1/2, 1/root 2, root 3 / 2, 1. cos values are the same in the reverse order. tan 0, 30, 45, 60 = 0, 1/root 3, 1, root 3 and tan 90 is not defined.

What are the three main trigonometric identities?

sin^2 theta + cos^2 theta = 1, 1 + tan^2 theta = sec^2 theta and 1 + cot^2 theta = cosec^2 theta.

What are complementary angle formulas?

sin(90 - theta) = cos theta, cos(90 - theta) = sin theta, tan(90 - theta) = cot theta and so on. They are used to simplify products like tan 15 x tan 75.

What is the angle of elevation and depression?

The angle of elevation is the angle between the horizontal and the line of sight when looking up at an object. The angle of depression is the angle between the horizontal and the line of sight when looking down.

How do I solve heights and distances questions?

Draw the right triangle, mark the angle and the known side, decide which ratio connects the known and unknown sides (usually tan) and solve.

What is the trick to remember sin values?

Write the numbers 0, 1, 2, 3, 4, take the square root of each and divide by 2. These are sin 0, 30, 45, 60 and 90. Cos values are the same list reversed.

What if the question gives sin theta + cos theta?

Square both sides to use sin^2 + cos^2 = 1 and find the value of sin theta x cos theta, which helps in other expressions.

How should I prepare trigonometry for SSC exams?

Learn the standard value table and identities by heart, practise one set a day, solve heights and distances questions with diagrams and then attempt previous year papers.