Trigonometry for RRB Group D

Trigonometry in RRB Group D stays at Class 10 level. Questions fall into four kinds: put in values from the table, simplify using an identity, use complementary angles (sin 35° ÷ cos 55°), and simple heights and distances.

Many candidates skip the topic because the word sounds hard. That is a mistake. Once you know one table and three identities, these are among the quickest marks in the maths section.

The six ratios

Take a right-angled triangle and one of its acute angles, θ. Name the sides from θ's point of view: the side facing θ is the opposite, the side next to θ (not the hypotenuse) is the adjacent, and the longest side is the hypotenuse.

Two links save time: tan θ = sin θ ÷ cos θ, and cot θ = cos θ ÷ sin θ.

The table of values

The other three rows are the reciprocals. For example, sec 60° = 2, cosec 30° = 2 and cot 60° = 1/√3.

The three identities

• sin²θ + cos²θ = 1
• sec²θ − tan²θ = 1, which is the same as 1 + tan²θ = sec²θ
• cosec²θ − cot²θ = 1, which is the same as 1 + cot²θ = cosec²θ

Questions often hide these. For example, 1 − sin²θ is just cos²θ, and sec²θ − 1 is just tan²θ.

Complementary angles

Two angles that add up to 90° are complementary. For them:

• 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

So sin 35° = cos 55°, and tan 20° × tan 70° = tan 20° × cot 20° = 1. When you see two angles you can't find in the table, check whether they add up to 90°.

Heights and distances

• The angle of elevation is measured upwards from the horizontal to an object above you.
• The angle of depression is measured downwards from the horizontal to an object below you.

Nearly every question here gives a horizontal distance and asks for a height (or the reverse). The ratio that links height and horizontal distance is tan. Use sin or cos only when the slanting length (a ladder, a kite string) is given or asked.

Worked examples

Example 1: Find 2 sin 30° + 3 tan 45° − cos 0°.

• 2 × 1/2 + 3 × 1 − 1 = 1 + 3 − 1 = 3.

Example 2: If tan θ = 5/12, find sin θ and cos θ.

• Opposite = 5, adjacent = 12. The hypotenuse is 13 (the 5, 12, 13 triplet).
• sin θ = 5/13; cos θ = 12/13.

Example 3: Find sin 35° ÷ cos 55°.

• 35 + 55 = 90, so sin 35° = cos 55°.
• The value is 1.

Example 4: Simplify (1 − sin²θ) × sec²θ.

• 1 − sin²θ = cos²θ, and sec²θ = 1 ÷ cos²θ.
• The product is 1.

Example 5: From a point 30√3 m from the foot of a tower, the angle of elevation of the top is 30°. Find the height of the tower.

• tan 30° = height ÷ 30√3, so 1/√3 = height ÷ 30√3.
• Height = 30√3 ÷ √3 = 30 m.

Example 6: A ladder leans against a wall and makes an angle of 60° with the ground. Its foot is 4 m from the wall. How long is the ladder?

• The ladder is the hypotenuse, and 4 m is the side next to the 60° angle. So use cos.
• cos 60° = 4 ÷ length, so 1/2 = 4 ÷ length, and the length is 8 m.

Common mistakes

• Swapping the values of 30° and 60°, for example writing sin 30° = √3/2.
• Using sin when you know the horizontal distance and want the height. Use tan.
• Treating sin²θ as sin(θ²). It means (sin θ)².
• Writing tan 90° = 1 or 0. It is not defined.
• Giving the answer 20√3 when 20√3 ÷ √3 simplifies to 20. Always simplify √3 ÷ √3.

Practice set

1. Find cos 60° + sin 30°.
2. Find tan 30° × tan 60°.
3. If sin θ = 8/17 and θ is acute, find cos θ.
4. A pole is 20√3 m high. From a point on the ground, the angle of elevation of its top is 60°. How far is the point from the foot of the pole?
5. Find the value of cosec²θ − cot²θ.
6. Find sin²30° + cos²60°.
7. Find cos 40° ÷ sin 50°.
8. A kite is flying on a string 100 m long. The string makes an angle of 30° with the ground. How high is the kite (assume the string is straight)?

Answers:

1. 1. 1/2 + 1/2.
2. 1. 1/√3 × √3.
3. 15/17. Opposite 8, hypotenuse 17, so the adjacent is 15 (8, 15, 17).
4. 20 m. tan 60° = 20√3 ÷ distance, so √3 = 20√3 ÷ distance.
5. 1. It is one of the three identities.
6. 1/2. 1/4 + 1/4.
7. 1. 40 + 50 = 90, so cos 40° = sin 50°.
8. 50 m. The string is the hypotenuse: sin 30° = height ÷ 100, so height = 100 × 1/2.

What to do next

• Write the table from memory every morning for a week, using the √0 to √4 trick.
• Solve ten value questions and ten identity questions from a question bank.
• Draw a figure for every heights question and mark which side is known and which is asked before choosing the ratio.
• Revise the right-triangle facts in geometry, then add the table to your maths formula sheet.