Trigonometry Notes for Competitive Exams

Trigonometry is a favourite area for SSC CGL, CHSL, CPO, CDS and NDA examiners because a few identities can generate many questions. In UPSC CSAT, trigonometry itself is rarely asked, but the idea of heights and distances helps in some direction and geometry problems. The word trigonometry means "measurement of triangles". At the competitive exam level, it deals with the ratios between sides of a right-angled triangle, the identities that connect those ratios, and their use in finding heights and distances of real objects like towers, poles and buildings.

Quick Facts: Trigonometry at a Glance

• Six ratios: sin, cos, tan, cosec, sec and cot.
• Key identity: sin²θ + cos²θ = 1.
• Standard angles: 0°, 30°, 45°, 60° and 90°; their values must be memorised.
• Complementary rule: sin(90° − θ) = cos θ; tan(90° − θ) = cot θ.
• Heights and distances: Uses tan θ = height ÷ base in most questions.

1. Trigonometric Ratios

In a right-angled triangle, for an acute angle θ, the side opposite θ is the perpendicular (P), the side next to θ is the base (B) and the longest side is the hypotenuse (H).

Memory aid: "Pandit Badri Prasad Har Har Bhole" gives P/H, B/H, P/B for sin, cos, tan. Also, tan θ = sin θ ÷ cos θ and cot θ = cos θ ÷ sin θ.

Example: If sin θ = 3/5, then P = 3, H = 5 and B = 4 (Pythagorean triplet). So cos θ = 4/5, tan θ = 3/4 and sec θ = 5/4.

2. Values at Standard Angles

Trick to remember sin values: Write 0, 1, 2, 3, 4; divide each by 4; take the square root. You get 0, 1/2, 1/√2, √3/2, 1. Cos values are the same list in reverse order.

Example: sin 30° + cos 60° + tan 45° = 1/2 + 1/2 + 1 = 2.

3. Fundamental Identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ, so sec²θ − tan²θ = 1 and (sec θ + tan θ)(sec θ − tan θ) = 1.
• 1 + cot²θ = cosec²θ, so (cosec θ + cot θ)(cosec θ − cot θ) = 1.

Example: If sec θ + tan θ = 3, find sec θ − tan θ. Since their product is 1, sec θ − tan θ = 1/3. Adding the two: 2 sec θ = 10/3, so sec θ = 5/3. This trick is asked very often in SSC exams.

Example: Simplify (1 − sin²θ) × sec²θ. Since 1 − sin²θ = cos²θ and sec²θ = 1/cos²θ, the answer is 1.

4. Complementary Angles

Two angles are complementary when they add up to 90°. Then:

• sin(90° − θ) = cos θ and cos(90° − θ) = sin θ
• tan(90° − θ) = cot θ and cot(90° − θ) = tan θ
• sec(90° − θ) = cosec θ and cosec(90° − θ) = sec θ

Example: tan 1° × tan 2° × tan 3° ... × tan 89°. Pair tan 1° with tan 89° = cot 1°, giving 1 each time. The middle term is tan 45° = 1. So the product = 1.

Example: sin²20° + sin²70° = sin²20° + cos²20° = 1.

Useful rule: If A + B = 90°, then sin A = cos B, tan A × tan B = 1, and sin²A + sin²B = 1.

5. Maximum and Minimum Values

• sin θ and cos θ always lie between −1 and 1.
• For a sin θ + b cos θ, maximum = √(a² + b²) and minimum = −√(a² + b²).
• For a sin²θ + b cos²θ (a > b), maximum = a and minimum = b.
• For a² tan²θ + b² cot²θ, minimum value = 2ab.
• For sin θ × cos θ, maximum = 1/2.

Example: Maximum value of 3 sin θ + 4 cos θ = √(9 + 16) = 5.

Example: Minimum value of 9 tan²θ + 4 cot²θ = 2 × 3 × 2 = 12.

6. Value-Putting in Trigonometry

Like algebra, many trigonometry questions with expressions can be solved by putting θ = 45° or θ = 0° (as long as nothing becomes undefined).

Example: Find (sin⁴θ − cos⁴θ) ÷ (sin²θ − cos²θ). At θ = 30°, sin² = 1/4 and cos² = 3/4. Numerator = 1/16 − 9/16 = −1/2; denominator = −1/2. Answer = 1. (Algebraically, it simplifies to sin²θ + cos²θ = 1.)

Avoid θ = 45° when the expression has sin θ − cos θ in the denominator, since that would be zero.

7. Heights and Distances

This is the practical application of trigonometry.

• Angle of elevation: The angle between the horizontal and the line of sight when you look up at an object.
• Angle of depression: The angle between the horizontal and the line of sight when you look down at an object.
• The angle of elevation from point A to B equals the angle of depression from B to A (alternate angles).

Standard Ratios to Remember

Solved Example 1: The angle of elevation of the top of a tower from a point 30 m away from its foot is 60°. Height = 30 × tan 60° = 30√3 m (about 51.96 m).

Solved Example 2: From the top of a 50 m cliff, the angle of depression of a boat is 45°. Distance of the boat from the foot of the cliff = 50 ÷ tan 45° = 50 m.

Solved Example 3: A man walks towards a tower. The angle of elevation of its top changes from 30° to 60° after walking 40 m. Find the height. Let height = h. From the 60° point, base = h/√3; from the 30° point, base = h√3. Difference = h√3 − h/√3 = 2h/√3 = 40, so h = 20√3 m.

Shortcut: When elevation changes from 30° to 60° while moving a distance d towards the object, the height is always (√3/2) × d.

Common Mistakes to Avoid

1. Mixing up which side is the base and which is the perpendicular for the given angle.
2. Writing tan 90° = 0 instead of "not defined".
3. Forgetting that the angle of depression is measured from the horizontal, not the vertical.
4. Using value-putting at an angle that makes the expression undefined.
5. Not rationalising answers like 30/√3 = 10√3 to match options.

How to Prepare Trigonometry

• Step 1: Memorise the value table until you can write it in 30 seconds.
• Step 2: Practise 30 identity-based questions, especially sec θ ± tan θ and cosec θ ± cot θ types.
• Step 3: Learn complementary angle and maximum-minimum formulas with 20 questions.
• Step 4: Solve 25 heights and distances questions using the standard ratio table.
• Step 5: Attempt previous year SSC CGL and CDS questions in timed sets of 10.

Trigonometry is mostly a memory-plus-pattern chapter. Learn the table and the three identities well, and you will find that most questions reduce to one or two lines of working.

Practice Questions with Answers

1. If cos θ = 5/13, find tan θ. Answer: B = 5, H = 13, so P = 12 and tan θ = 12/5.
2. Find the value of 2 sin 30° cos 30°. Answer: 2 × 1/2 × √3/2 = √3/2, which is also sin 60°.
3. If cosec θ − cot θ = 1/4, find cosec θ + cot θ. Answer: 4, since their product is 1.
4. Find cos²25° + cos²65°. Answer: cos²25° + sin²25° = 1.
5. A kite is flying on a 100 m string making 30° with the ground. Find its height. Answer: 100 × sin 30° = 50 m.
6. Find the minimum value of 4 sec²θ + 9 cos²θ. Answer: By AM-GM, minimum = 2√(4 × 9) = 12.

Solve these first without looking at the answers. If you get at least five right within eight minutes, you are ready for exam-level mixed sets.

Related Notes

• Mensuration Notes: Area, Volume and Perimeter
• Number System Notes
• Ratio and Proportion Notes
• All Quantitative Aptitude Notes

Frequently Asked Questions

What are the six trigonometric ratios?

Sine, cosine, tangent, cosecant, secant and cotangent. The last three are reciprocals of the first three.

What is the most important trigonometric identity?

sin²θ + cos²θ = 1. The other two identities, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ, come from it.

What is the value of tan 90°?

tan 90° is not defined because cos 90° = 0 and division by zero is not possible.

What is the value of sin 30° and cos 60°?

Both are equal to 1/2.

What is the complementary angle rule?

For angles that add up to 90°, sin(90° − θ) = cos θ, tan(90° − θ) = cot θ and sec(90° − θ) = cosec θ.

If sec θ + tan θ = x, what is sec θ − tan θ?

It is 1/x, because (sec θ + tan θ)(sec θ − tan θ) = 1.

What is the maximum value of a sin θ + b cos θ?

The maximum value is √(a² + b²) and the minimum is −√(a² + b²).

What is angle of elevation?

It is the angle between the horizontal line and the line of sight when looking up at an object.

What is angle of depression?

It is the angle between the horizontal line and the line of sight when looking down at an object.

What is the height to base ratio at 30°?

At 30°, height : base : hypotenuse = 1 : √3 : 2.

Is trigonometry asked in UPSC CSAT?

Direct trigonometry questions are rare in CSAT. It is much more important for SSC, CDS and NDA exams.

How do I remember the trigonometry table easily?

For sin, write 0 to 4, divide by 4 and take the square root. For cos, reverse the sin values. Tan is sin divided by cos.