Trigonometry: Ratios, Identities, Heights and Distances

What to remember

• In a right triangle, sin = opposite/hypotenuse, cos = adjacent/hypotenuse and tan = opposite/adjacent; cosec, sec and cot are their reciprocals.
• Basic identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ; complementary angles give sin(90° − θ) = cos θ.
• Heights and distances use the angle of elevation or depression with a right triangle, and usually tan, sin or cos of 30°, 45° or 60°.

Trigonometric ratios

Take a right triangle with 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).

Also tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. The ratios depend only on the angle, not on the size of the triangle, because right triangles with the same acute angle are similar. For an acute angle, sin θ and cos θ lie between 0 and 1, while sec θ and cosec θ are at least 1. The ratio sin θ is one symbol for the ratio; it does not mean sin × θ.

If one ratio is known, the others follow from a right triangle. Example: sin A = 3/5 gives sides 3, 4, 5, so cos A = 4/5 and tan A = 3/4.

Standard values

A memory aid for sine: the values √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. Cosine is the same list in reverse order. As θ goes from 0° to 90°, sin and tan increase, while cos decreases.

Complementary angles

Examples: sin 35° / cos 55° = 1, because cos 55° = sin 35°. tan 15° × tan 75° = tan 15° × cot 15° = 1.

Identities

• 1. sin²θ + cos²θ = 1 (true for every angle).
• 2. 1 + tan²θ = sec²θ (divide the first by cos²θ).
• 3. 1 + cot²θ = cosec²θ (divide the first by sin²θ).
• 4. sin θ × cosec θ = 1, cos θ × sec θ = 1, tan θ × cot θ = 1.

Useful consequences: sec²θ − tan²θ = 1; cosec²θ − cot²θ = 1; 1 − sin²θ = cos²θ. To prove an identity, work on one side (usually the more complex one) and convert everything to sin and cos. Do not move terms across the equals sign as if the result were already true.

Worked examples:

• (sin θ + cos θ)² + (sin θ − cos θ)² = 2(sin²θ + cos²θ) = 2.
• (1 − sin²θ) sec²θ = cos²θ × sec²θ = 1.
• If sec θ = 5/4, then tan²θ = 25/16 − 1 = 9/16, so tan θ = 3/4.
• sin 60° cos 30° + cos 60° sin 30° = 3/4 + 1/4 = 1.
• 2 sin 30° cos 30° = 2 × 1/2 × √3/2 = √3/2.

Proving an identity: a model

To prove (1 + tan²θ) cos²θ = 1, take the left side. Since 1 + tan²θ = sec²θ, the left side becomes sec²θ × cos²θ. Because sec θ = 1/cos θ, this equals 1, which is the right side. Another model is to prove sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2 cosec θ: combine the fractions to get (sin²θ + (1 + cos θ)²)/(sin θ (1 + cos θ)). The numerator is sin²θ + 1 + 2 cos θ + cos²θ = 2 + 2 cos θ = 2(1 + cos θ). Cancelling (1 + cos θ) leaves 2/sin θ = 2 cosec θ.

Other special results: sin θ increases with θ in the first quadrant, so sin 20° < sin 40°; cos θ decreases, so cos 20° > cos 40°. An angle of 0° gives sin 0° = 0, cos 0° = 1 and tan 0° = 0. Ratios of 22.5° or 15° are not in the standard table and are given in the question when needed.

Heights and distances

The line of sight is the line from the eye to the object. The angle of elevation is the angle between the line of sight and the horizontal when the object is above the observer's eye. The angle of depression is the angle between the line of sight and the horizontal when the object is below. The angle of elevation of the object from the observer equals the angle of depression of the observer from the object, because they are alternate angles.

Method: (1) draw a neat figure; (2) mark the right triangle; (3) choose the ratio that links the known and unknown sides; (4) solve. For height h and horizontal distance d: tan θ = h/d. For a ladder of length l leaning on a wall: sin θ = height/l and cos θ = foot distance/l. Ignore the observer's height unless it is given.

Worked examples:

• 1. A tower is seen at 30° from 30 m away: h = 30 tan 30° = 10√3 m.
• 2. A pole of 10 m has a shadow of 10 m: tan θ = 1, so θ = 45°.
• 3. A 20 m ladder reaches 10 m up the wall: sin θ = 1/2, so θ = 30°.
• 4. From a 50 m tower a boat is seen at a depression of 45°: the boat is 50 m from the foot.
• 5. A 60 m building at an elevation of 60°: d = 60/√3 = 20√3 m.

When two angles of elevation are given from two points in line with the tower, set up two equations in h and d and subtract. For example, with tower height h and elevation angles 30° and 60° at distances d + x and d, h = (d + x)/√3 and h = d√3.

Classroom angle

Begin with similar right triangles drawn on squared paper and let learners measure the ratios for a fixed angle; this shows why the ratio depends only on the angle. Use a clinometer made from a protractor, a straw and a thread to measure real heights in the school yard. Misconceptions: reading "sin θ" as sin × θ, mixing opposite and adjacent sides, using elevation as the angle with the vertical, and forgetting that tan 90° is undefined.

Exam traps

• sin θ is not sin × θ; it is a single ratio.
• tan 90° and sec 90° are undefined; cot 0° and cosec 0° are undefined.
• sin 30° = cos 60° = 1/2, but tan 30° = 1/√3 and tan 60° = √3.
• tan (90° − θ) = cot θ, not tan θ.
• sin²θ + cos²θ = 1 holds for all angles, not just acute ones.
• sec θ for an acute angle is never less than 1.
• The angle of elevation is measured from the horizontal, not from the vertical.
• The sine and cosine of an angle can never be greater than 1.

One-liners

• 1. sin = P/H, cos = B/H, tan = P/B.
• 2. sin 30° = 1/2 and cos 60° = 1/2.
• 3. sin 45° = cos 45° = 1/√2.
• 4. tan 45° = 1.
• 5. tan 60° = √3 and tan 30° = 1/√3.
• 6. sin²θ + cos²θ = 1.
• 7. 1 + tan²θ = sec²θ.
• 8. 1 + cot²θ = cosec²θ.
• 9. sin (90° − θ) = cos θ.
• 10. tan θ × cot θ = 1.
• 11. The angle of elevation equals the angle of depression from the other end.
• 12. Height = distance × tan (angle of elevation).

Practice questions

1. In a right triangle, sin θ is defined as

   1. hypotenuse / opposite side
   2. opposite side / adjacent side
   3. adjacent side / hypotenuse
   4. opposite side / hypotenuse
   Answer

   D. opposite side / hypotenuse

   Sine is perpendicular over hypotenuse.

2. tan θ is equal to

   1. sin θ × cos θ
   2. sin θ / cos θ
   3. cos θ / sin θ
   4. 1 / sin θ
   Answer

   B. sin θ / cos θ

   tan θ = opposite/adjacent = sin θ / cos θ.

3. The reciprocal of sin θ is

   1. sec θ
   2. tan θ
   3. cot θ
   4. cosec θ
   Answer

   D. cosec θ

   cosec θ = 1/sin θ.

4. Which identity is correct for all angles?

   1. sin²θ + cos²θ = 1
   2. tan²θ + cot²θ = 1
   3. sin²θ − cos²θ = 1
   4. sin θ + cos θ = 1
   Answer

   A. sin²θ + cos²θ = 1

   This follows from the Pythagoras theorem.

5. 1 + tan²θ equals

   1. cos²θ
   2. cosec²θ
   3. sec²θ
   4. cot²θ
   Answer

   C. sec²θ

   Divide sin²θ + cos²θ = 1 by cos²θ.

6. 1 + cot²θ equals

   1. tan²θ
   2. sin²θ
   3. sec²θ
   4. cosec²θ
   Answer

   D. cosec²θ

   Divide sin²θ + cos²θ = 1 by sin²θ.

7. The value of sin 30° is

   1. 1/2
   2. √3/2
   3. 1/√2
   4. 1
   Answer

   A. 1/2

   A standard value.

8. The value of cos 60° is

   1. √3/2
   2. 1/2
   3. 1/√2
   4. 0
   Answer

   B. 1/2

   cos 60° = sin 30° = 1/2.

9. The value of tan 45° is

   1. 0
   2. √3
   3. 1
   4. 1/√3
   Answer

   C. 1

   Opposite equals adjacent at 45°.

10. The value of tan 60° is

    1. √3
    2. 1/√3
    3. 1
    4. 2
    Answer

    A. √3

    tan 60° = sin 60°/cos 60° = √3.

11. The value of sin 45° is

    1. 1/2
    2. √3/2
    3. 1
    4. 1/√2
    Answer

    D. 1/√2

    A standard value.

12. The value of sec 60° is

    1. 1/2
    2. 2
    3. √3
    4. 2/√3
    Answer

    B. 2

    sec 60° = 1/cos 60° = 2.

13. Which of the following is undefined?

    1. cot 45°
    2. sin 90°
    3. tan 90°
    4. tan 45°
    Answer

    C. tan 90°

    cos 90° = 0, so tan 90° has zero in the denominator.

14. Which value is impossible for sin θ?

    1. 0.5
    2. 1
    3. 0
    4. 1.5
    Answer

    D. 1.5

    The sine of any angle lies between −1 and 1.

15. sin (90° − θ) equals

    1. cos θ
    2. tan θ
    3. sin θ
    4. sec θ
    Answer

    A. cos θ

    The sine of an angle equals the cosine of its complement.

16. If sin A = 3/5 and A is acute, then tan A is

    1. 4/5
    2. 3/5
    3. 3/4
    4. 4/3
    Answer

    C. 3/4

    The third side is 4, so tan A = 3/4.

17. If cos A = 5/13 and A is acute, then sin A is

    1. 8/13
    2. 12/13
    3. 13/12
    4. 5/12
    Answer

    B. 12/13

    sin A = √(1 − 25/169) = 12/13.

18. The value of sin 30° + cos 60° is

    1. 1
    2. 1/2
    3. √3
    4. 3/2
    Answer

    A. 1

    1/2 + 1/2 = 1.

19. The value of tan 30° × tan 60° is

    1. √3
    2. 1/3
    3. 3
    4. 1
    Answer

    D. 1

    (1/√3) × √3 = 1.

20. The value of 2 sin 30° cos 30° is

    1. 1/2
    2. 3/4
    3. √3/2
    4. √3
    Answer

    C. √3/2

    2 × 1/2 × √3/2 = √3/2.

21. The value of sin 60° cos 30° + cos 60° sin 30° is

    1. √3/2
    2. 1
    3. 1/2
    4. 3/4
    Answer

    B. 1

    3/4 + 1/4 = 1.

22. The value of (sin θ + cos θ)² + (sin θ − cos θ)² is

    1. 1
    2. 0
    3. 4 sin θ cos θ
    4. 2
    Answer

    D. 2

    The cross terms cancel; 2(sin²θ + cos²θ) = 2.

23. The expression (1 − sin²θ) sec²θ simplifies to

    1. 1
    2. 0
    3. tan²θ
    4. sin²θ
    Answer

    A. 1

    cos²θ × sec²θ = 1.

24. The value of tan 15° × tan 75° is

    1. 0
    2. √3
    3. 1
    4. 1/√3
    Answer

    C. 1

    tan 75° = cot 15°, so the product is 1.

25. The value of sin 35° / cos 55° is

    1. 0
    2. tan 35°
    3. 1
    4. 1/2
    Answer

    C. 1

    cos 55° = sin 35°.

26. If sin θ = cos θ for an acute angle θ, then θ is

    1. 60°
    2. 45°
    3. 30°
    4. 90°
    Answer

    B. 45°

    Equal only at 45°.

27. If tan θ = √3 for an acute angle θ, then θ is

    1. 60°
    2. 90°
    3. 30°
    4. 45°
    Answer

    A. 60°

    tan 60° = √3.

28. If sec θ = 5/4 and θ is acute, then tan θ is

    1. 3/5
    2. 3/4
    3. 5/3
    4. 4/3
    Answer

    B. 3/4

    tan²θ = sec²θ − 1 = 9/16.

29. A pole 10 m high casts a shadow 10 m long. The angle of elevation of the sun is

    1. 60°
    2. 90°
    3. 30°
    4. 45°
    Answer

    D. 45°

    tan θ = 10/10 = 1.

30. The angle of elevation of the top of a tower from a point 30 m from its foot is 30°. The height of the tower is

    1. 10√3 m
    2. 30√3 m
    3. 30 m
    4. 15 m
    Answer

    A. 10√3 m

    h = 30 tan 30° = 30/√3 = 10√3.

31. A 10 m ladder leans on a wall making 60° with the ground. The height it reaches is

    1. 10√3 m
    2. 5√3 m
    3. 5 m
    4. 20/√3 m
    Answer

    B. 5√3 m

    h = 10 sin 60° = 5√3.

32. A 20 m ladder reaches 10 m up a wall. The angle it makes with the ground is

    1. 60°
    2. 15°
    3. 30°
    4. 45°
    Answer

    C. 30°

    sin θ = 10/20 = 1/2.

33. A kite string is 100 m long and makes 30° with the ground; the string is taut. The kite's height is

    1. 100/√3 m
    2. 50√3 m
    3. 100 m
    4. 50 m
    Answer

    D. 50 m

    h = 100 sin 30° = 50.

34. From the top of a 50 m tower the angle of depression of a boat is 45°. The boat's distance from the tower's foot is

    1. 25 m
    2. 50√2 m
    3. 50 m
    4. 50√3 m
    Answer

    C. 50 m

    tan 45° = 50/d, so d = 50.

35. A 60 m building is viewed at an elevation of 60° from a point on the ground. The distance from the foot is

    1. 60√3 m
    2. 20√3 m
    3. 30 m
    4. 60 m
    Answer

    B. 20√3 m

    d = 60/tan 60° = 60/√3 = 20√3.

36. The angle of depression of an object from an observer equals

    1. the angle of elevation of the observer from the object
    2. zero
    3. the supplement of that angle
    4. twice that angle
    Answer

    A. the angle of elevation of the observer from the object

    Alternate angles between parallel horizontal lines are equal.

37. Consider the statements: 1. sin θ and cos θ never exceed 1 for any angle. 2. sec θ can be less than 1 for an acute angle. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    For an acute angle sec θ ≥ 1.

38. Consider the statements: 1. tan θ = sin θ / cos θ. 2. cot θ = cos θ / sin θ. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard definitions.

39. Consider the statements: 1. sin (90° − θ) = cos θ. 2. tan (90° − θ) = tan θ. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    tan (90° − θ) = cot θ.

40. Consider the statements: 1. cos 90° = 1. 2. sin 90° = 1. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    cos 90° = 0.

41. Consider the statements: 1. tan 90° = 1. 2. sec 90° is defined. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    Both tan 90° and sec 90° are undefined.

42. Which pair is correctly matched?

    1. cot 45° : 0
    2. tan 30° : √3
    3. sec 60° : 1/2
    4. cosec 30° : 2
    Answer

    D. cosec 30° : 2

    cosec 30° = 1/sin 30° = 2.

43. Which identity pair is correctly matched?

    1. sec²θ − 1 : cot²θ
    2. 1 − sin²θ : sin²θ
    3. 1 + tan²θ : cosec²θ
    4. 1 + cot²θ : cosec²θ
    Answer

    D. 1 + cot²θ : cosec²θ

    The others are wrong: sec²θ − 1 = tan²θ and 1 − sin²θ = cos²θ.

44. A learner writes 'sin θ = sin × θ'. The best teacher response is to explain that

    1. sin is a number that is always 1/2
    2. θ should be replaced by 30° always
    3. sin θ is a single ratio that depends on the angle, not a product
    4. the learner is correct
    Answer

    C. sin θ is a single ratio that depends on the angle, not a product

    Trig ratios are functions of the angle, not multiplications.

45. The best way to introduce trigonometric ratios in Class X is to

    1. give the table of values first
    2. state the identities without examples
    3. draw several similar right triangles and show that the ratios of sides depend only on the angle
    4. teach only heights and distances
    Answer

    C. draw several similar right triangles and show that the ratios of sides depend only on the angle

    Similar triangles show why ratios are the same for a given angle.