Introduction to trigonometry: Class 10 Mathematics Notes, Short Questions and Numericals

Angles in degrees and radians, arc length and sector area, the six trigonometric ratios, values for 30°, 45° and 60°, fundamental identities, and simple problems on heights and distances.

Key points

Trigonometry studies the relations between the sides and angles of triangles. An angle is formed by rotating a ray about its starting point. The angle is positive for anticlockwise rotation and negative for clockwise rotation. The common units of angle are the degree and the radian.

A degree is 1/360 of a complete turn. One degree is divided into 60 minutes and one minute into 60 seconds, so 1° = 60′ and 1′ = 60″. A radian is the angle at the centre of a circle subtended by an arc whose length equals the radius. A complete turn is 2π radians, so π radians = 180°, and 1 radian ≈ 57.3°.

Changing units:

Degrees to radians: multiply by π/180.
Radians to degrees: multiply by 180/π.
For example 90° = π/2 rad, 60° = π/3 rad, 45° = π/4 rad, 30° = π/6 rad.

If a circle of radius r has an arc of length s that subtends an angle θ radians at the centre, then s = rθ. The area of the sector is A = ½ r² θ = ½ r s. The angle θ must be in radians in these formulas.

In a right angled triangle, the side opposite the right angle is the hypotenuse. For an acute angle θ, the side facing θ is the opposite (perpendicular) and the other side touching θ is the adjacent (base). The Pythagorean theorem states that (hypotenuse)² = (opposite)² + (adjacent)².

The six trigonometric ratios for an acute angle θ:

sin θ = opposite / hypotenuse.
cos θ = adjacent / hypotenuse.
tan θ = opposite / adjacent.
cosec θ = 1 / sin θ = hypotenuse / opposite.
sec θ = 1 / cos θ = hypotenuse / adjacent.
cot θ = 1 / tan θ = adjacent / opposite.

Also tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. The ratios depend only on the angle and not on the size of the triangle, because similar triangles have equal ratios of sides.

The values of the ratios for the special angles come from a 45° right isosceles triangle with sides 1, 1, √2 and a 30°, 60° triangle with sides 1, √3, 2. They should be memorised.

Values of the ratios for special angles (30°, 45°, 60°):

sin: 1/2, 1/√2, √3/2.
cos: √3/2, 1/√2, 1/2.
tan: 1/√3, 1, √3.

The fundamental identities are true for every angle: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. The first comes from the Pythagorean theorem by dividing by the hypotenuse squared. The other two are found by dividing the first by cos²θ and sin²θ.

Angles of elevation and depression give applications. The angle of elevation is the angle above the horizontal when we look up at an object. The angle of depression is the angle below the horizontal when we look down. Draw a right triangle, mark the known side and angle, and use sin, cos or tan to find the unknown height or distance.

Study tip: Learn the table of special angle values by drawing the 45° and the 30°, 60° triangles, so you can rebuild them in a minute. In identity proofs work on the left side only, change everything to sin and cos, and stop when it matches the right side. In word problems always begin with a neat labelled diagram.

Definitions and flashcards

Say the meaning first, then open the card to check.

Degree

The unit of angle equal to 1/360 of a full turn.

Radian

The angle at the centre of a circle subtended by an arc equal to the radius.

Hypotenuse

The side opposite the right angle in a right triangle.

Sine

The ratio of the opposite side to the hypotenuse.

Cosine

The ratio of the adjacent side to the hypotenuse.

Tangent

The ratio of the opposite side to the adjacent side.

Identity

An equation that is true for all values of the angle.

Angle of elevation

The angle between the horizontal and the line of sight to an object above.

Formula sheet

Name	Formula	What it means
Radian measure	π rad = 180°	Basis for converting units.
Arc length	s = rθ	θ in radians.
Sector area	A = ½ r² θ	θ in radians.
Ratios	sin θ = p/h, cos θ = b/h, tan θ = p/b	p perpendicular, b base, h hypotenuse.
Main identity	sin²θ + cos²θ = 1	True for every angle.
Other identities	1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ	Derived from the main identity.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Define one radian. (2 marks)

One radian is the angle at the centre of a circle which is subtended by an arc whose length is equal to the radius of the circle.

Convert 135° to radians. (2 marks)

Multiply by π/180. 135 × π/180 = 3π/4 radians.

State the reciprocal relations among the trigonometric ratios. (2 marks)

cosec θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ.

Prove that sin²θ + cos²θ = 1. (3 marks)

In a right triangle with perpendicular p, base b and hypotenuse h, sin θ = p/h and cos θ = b/h. Then sin²θ + cos²θ = (p² + b²)/h². By the Pythagorean theorem p² + b² = h², so the expression equals h²/h² = 1.

Prove that 1 + tan²θ = sec²θ. (3 marks)

Start with sin²θ + cos²θ = 1. Divide every term by cos²θ. This gives sin²θ/cos²θ + 1 = 1/cos²θ, that is tan²θ + 1 = sec²θ.

Find the values of sin 60°, cos 60° and tan 60° using a triangle. (5 marks)

Take an equilateral triangle of side 2 and draw an altitude from one vertex. It splits the triangle into two right triangles with angles 30°, 60° and 90°. The base is 1, the hypotenuse is 2, and the altitude is √(4 − 1) = √3. For the angle 60°, the opposite side is √3 and the adjacent side is 1. So sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3/1 = √3.

Numericals with worked steps

Convert 72° to radians and 3π/4 radians to degrees.

72 × π/180 = 72π/180 = 2π/5 rad.
(3π/4) × 180/π = 3 × 45 = 135.

Answer: 72° = 2π/5 rad and 3π/4 rad = 135°

A circle has radius 7 cm. Find the arc length and sector area for a central angle of 60°. Use π = 22/7.

θ = 60 × π/180 = π/3 rad.
s = rθ = 7 × π/3 = 7 × 22/(7 × 3) = 22/3 = 7.33 cm.
A = ½ r² θ = ½ × 49 × π/3 = 49 × 22/(6 × 7) = 1078/42 = 25.67 cm².

Answer: Arc length ≈ 7.33 cm, sector area ≈ 25.67 cm²

In a right triangle, the side opposite θ is 5 cm and the adjacent side is 12 cm. Find sin θ, cos θ and tan θ.

Hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13 cm.
sin θ = 5/13.
cos θ = 12/13.
tan θ = 5/12.

Answer: sin θ = 5/13, cos θ = 12/13, tan θ = 5/12

If sin θ = 3/5 and θ is acute, find cos θ, tan θ and cosec θ.

cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25.
cos θ = 4/5 (positive, since θ is acute).
tan θ = sin θ / cos θ = (3/5) ÷ (4/5) = 3/4.
cosec θ = 1 / sin θ = 5/3.

Answer: cos θ = 4/5, tan θ = 3/4, cosec θ = 5/3

A ladder 10 m long leans against a wall and makes an angle of 60° with the ground. Find how high it reaches and how far its foot is from the wall.

Height = 10 sin 60° = 10 × √3/2 = 5√3 ≈ 5 × 1.732 = 8.66 m.
Distance of foot = 10 cos 60° = 10 × 1/2 = 5 m.
Check: 8.66² + 5² ≈ 75 + 25 = 100 = 10².

Answer: Height ≈ 8.66 m and distance of foot = 5 m

Common mistakes

Using a calculator in degree mode for a radian question, or the other way round.
Putting θ in degrees into s = rθ. The angle must be in radians.
Mixing up opposite and adjacent sides, which swaps sin and cos.
Writing sin²θ as sin θ² or forgetting that it means (sin θ)².
Learning the values for 30° and 60° the wrong way round.

Practise this chapter

Practice questions and timed test Printable worksheet