Introduction to Trigonometry · Learner pages

Lesson 1 · Build ratios from a right triangle

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

Name or initials: ____________   Date: ____________

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.

1. For acute angle θ, opposite = 3, adjacent = 4, hypotenuse = 5. Find sin θ.

   1. 3/5 · 2. 4/5 · 3. 3/4

2. For acute angle θ, opposite = 3, adjacent = 4, hypotenuse = 5. Find cos θ.

   1. 3/5 · 2. 3/4 · 3. 4/5

3. For acute angle θ, opposite = 5, adjacent = 12, hypotenuse = 13. Find tan θ.

   1. 12/13 · 2. 5/12 · 3. 5/13

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.

Name or initials: ____________   Date: ____________

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.

1. What is sin 30°?

   1. 1/2 · 2. √3/2 · 3. 1

2. What is cos 60°?

   1. 0 · 2. √3/2 · 3. 1/2

3. What is tan 45°?

   1. √3 · 2. 1 · 3. 1/2

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.

Name or initials: ____________   Date: ____________

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.

1. θ is acute and sin θ = 3/5. What is cos θ?

   1. 4/5 · 2. 3/5 · 3. 1

2. θ is acute and sin θ = 5/13. What is cos θ?

   1. 5/13 · 2. 1 · 3. 12/13

3. θ is acute and sin θ = 8/17. What is cos θ?

   1. 1 · 2. 15/17 · 3. 8/17

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

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. For acute angle θ, opposite = 7, adjacent = 24, hypotenuse = 25. Find cos θ.

   7/25 / 7/24 / 24/25

2. For acute angle θ, opposite = 20, adjacent = 21, hypotenuse = 29. Find tan θ.

   21/29 / 20/21 / 20/29

3. What is cos 30°?

   1/2 / 1 / √3/2

4. What is sin 45°?

   1 / 1/√2 / 1/2

5. θ is acute and sin θ = 4/5. What is cos θ?

   4/5 / 1 / 3/5

6. θ is acute and sin θ = 12/13. What is cos θ?

   1 / 5/13 / 12/13

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. For acute angle θ, opposite = 9, adjacent = 12, hypotenuse = 15. Find sin θ.

   3/5 / 4/5 / 3/4

2. For acute angle θ, opposite = 12, adjacent = 16, hypotenuse = 20. Find cos θ.

   3/5 / 3/4 / 4/5

3. What is tan 30°?

   1/√3 / √3 / 1

4. What is tan 60°?

   1/√3 / 1 / √3

5. θ is acute and sin θ = 15/17. What is cos θ?

   8/17 / 15/17 / 1

6. θ is acute and sin θ = 24/25. What is cos θ?

   24/25 / 1 / 7/25