HelioLearnLearn · practise · understand

GRADE 10 · Mathematics · LEARNING COMPANION

Introduction to Trigonometry

Three guided lessons with explanations, visual steps, practice and a final check.

Original learning companion, not full textbook exercise solutions. Covers right-triangle ratios, standard acute-angle values and basic identities. It does not exhaust all identity proofs or textbook exercises.

Read all lessons

These explanations remain available without JavaScript. The separate learner and solution pages can be printed on A4.

LESSON 1

Build ratios from a right triangle

Choose sine, cosine and tangent relative to an acute angle.

The hypotenuse is opposite the right angle and is the longest side.

  1. Choose the reference angle

    In right triangle ABC with right angle B, use θ at A. BC is opposite θ, AB is adjacent and AC is the hypotenuse.

  2. Insert labelled lengths

    Let BC = 3, AB = 4 and AC = 5. These satisfy 3² + 4² = 5².

  3. Form the three ratios

    sin θ = 3/5, cos θ = 4/5 and tan θ = 3/4. The ratios have no length unit.

  4. Change angle carefully

    Using the other acute angle swaps opposite and adjacent. The hypotenuse stays the same. Similar triangles give the same ratios for the same angle.

Worked example: A right triangle has opposite side 5 and hypotenuse 13. Adjacent = √(13² − 5²) = 12. Thus sin θ = 5/13, cos θ = 12/13 and tan θ = 5/12.

Watch for this: Opposite and adjacent depend on the chosen acute angle. The hypotenuse does not.

Discuss: Draw two similar right triangles, one twice as large. Show that their sine ratios for matching angles are equal.

LESSON 2

Derive familiar angle values

Connect 30°, 45° and 60° values to special triangles.

An equilateral triangle has three 60° angles; bisecting it creates right triangles.

  1. Build a 45° triangle

    An isosceles right triangle with legs 1 and 1 has hypotenuse √2. Thus sin 45° = cos 45° = 1/√2 and tan 45° = 1.

  2. Split an equilateral triangle

    An equilateral triangle of side 2 splits into a right triangle with sides 1, √3 and 2.

  3. Read 30° and 60° ratios

    The side opposite 30° is 1, so sin 30° = 1/2 and cos 30° = √3/2. For 60°, the opposite side is √3.

  4. Keep undefined values distinct

    At the limiting angle 90°, cos 90° = 0, so tan 90° is undefined because it would divide by zero.

Worked example: Evaluate 2sin 30° + cos 60°: 2 × 1/2 + 1/2 = 3/2. This differs from sin 60°; trigonometric functions do not distribute over angle addition.

Watch for this: tan 90° is undefined, not zero. Exact surds avoid rounding errors in later steps.

Discuss: Make a table of sin, cos and tan at 0°, 30°, 45°, 60° and 90°. Label the undefined entry explicitly.

LESSON 3

Use identities with conditions

Derive sin²θ + cos²θ = 1 and use it to find a missing ratio.

Pythagoras says opposite² + adjacent² = hypotenuse².

  1. Divide Pythagoras by the hypotenuse squared

    o² + a² = h² becomes (o/h)² + (a/h)² = 1.

  2. Recognise the ratios

    Therefore sin²θ + cos²θ = 1. Dividing by cos²θ where it is nonzero gives 1 + tan²θ = sec²θ.

  3. Find a missing acute-angle ratio

    If sin θ = 3/5, then cos²θ = 1 − 9/25 = 16/25. For acute θ, cos θ = 4/5.

  4. Check the denominator conditions

    Dividing by sin²θ where nonzero gives 1 + cot²θ = cosec²θ. Do not use a ratio at an angle where its denominator is zero.

Worked example: For acute θ with cos θ = 12/13, sin θ = √(1 − 144/169) = 5/13. Then tan θ = (5/13)/(12/13) = 5/12.

Watch for this: sin²θ means (sin θ)²; it is not sin(θ²). Taking a square root may require a sign condition.

Discuss: Explain why 1 − sin²θ equals cos²θ, while 1 − sin θ does not generally equal cos θ.

Continue on paper

Open A4 learner pagesOpen separate answer guide
Sources and coverage

Original learning companion, not full textbook exercise solutions. Covers right-triangle ratios, standard acute-angle values and basic identities. It does not exhaust all identity proofs or textbook exercises.

Original HelioLearn teaching and questions. Source-aligned companion; not an official NCERT/CBSE publication or full textbook solution bank. Independent teacher review and narrated video are pending.