Trigonometry and Heights and Distances

What to remember

• In a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Their reciprocals are cosec, sec and cot.
• Learn the standard values at 0°, 30°, 45°, 60° and 90°, and the three identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
• In heights and distances, draw the right triangle, mark the angle of elevation or depression from the horizontal, and use tan θ = height/distance.

Trigonometric ratios

Take a right triangle with an acute angle θ. The side facing θ is the opposite (perpendicular), the side next to θ is the adjacent (base), and the longest side is the hypotenuse.

The values of sin θ and cos θ for an acute angle lie between 0 and 1. The values of sec θ and cosec θ are at least 1. The value of tan θ can be any positive number.

Worked example. If sin θ = 3/5, the opposite is 3 and the hypotenuse is 5. The adjacent is √(25 − 9) = 4. So cos θ = 4/5 and tan θ = 3/4.

Another example. If tan θ = 5/12, the hypotenuse is √(25 + 144) = 13, so sin θ = 5/13. If sec θ = 5/3, then tan θ = √(25/9 − 1) = 4/3. If cos θ = 5/13, then sin θ = 12/13 and cosec θ = 13/12.

Standard values

Learn this table. As θ goes from 0° to 90°, sin θ increases from 0 to 1 and cos θ decreases from 1 to 0.

A memory aid for the sine row is √0/2, √1/2, √2/2, √3/2, √4/2. The cosine row is the same list in reverse.

Worked examples.

• sin 30° cos 60° + cos 30° sin 60° = 1/4 + 3/4 = 1.
• cos²30° − sin²30° = 3/4 − 1/4 = 1/2.
• 4 sin 30° cos 60° = 4 × 1/2 × 1/2 = 1.

Complementary angles and identities

For an acute angle θ:

• 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 25° / cos 65° = 1, because cos 65° = sin 25°. Also tan 10° × tan 80° = tan 10° × cot 10° = 1.

The three basic identities are true for every angle where the expressions are defined.

Other useful results: sin θ × cosec θ = 1, cos θ × sec θ = 1 and tan θ × cot θ = 1. Also (1 − cos²θ) / sin²θ = 1. If sin θ = cos θ, then θ = 45°. If tan θ = √3, then θ = 60°.

Simplifying. To prove an identity, start with the more complicated side. Convert everything to sin and cos, then use sin²θ + cos²θ = 1.

Heights and distances

Some terms must be clear.

• Line of sight: the line from the eye of the observer to the object.
• Angle of elevation: the angle between the horizontal line and the line of sight when the object is above the horizontal. The observer looks up.
• Angle of depression: the angle between the horizontal line and the line of sight when the object is below the horizontal. The observer looks down.
• The angle of depression of an object from a point equals the angle of elevation of that point from the object. They are alternate angles between parallel lines.

Method.

• 1. Draw a clear figure with a vertical height and a horizontal ground line.
• 2. Mark the known angle from the horizontal and the known length.
• 3. Choose the ratio that links the known and unknown sides. For height h and distance d, tan θ = h/d. For a slanted length l (ladder, string, rope), sin θ = h/l and cos θ = d/l.

Useful standard triangles.

• 45°–45°–90° triangle: sides in the ratio 1 : 1 : √2.
• 30°–60°–90° triangle: sides in the ratio 1 : √3 : 2 (opposite 30°, 60° and 90°).

Worked examples in heights and distances

A tree broken by a storm. The top of a tree touches the ground at 10√3 m from the foot, making 30° with the ground. The standing part is h = 10√3 × tan 30° = 10 m. The broken part is the hypotenuse, 10 / sin 30° = 20 m. The original height is 10 + 20 = 30 m.

Two points on the same line. Suppose the elevations of the top of a tower from two points on one line through its foot, at distances a and b from the foot, are complementary. Then tan θ = h/a and tan(90° − θ) = h/b. Multiplying gives 1 = h²/(ab), so h = √(ab). For a = 4 m and b = 9 m, h = 6 m.

Observer on a tower. If the tower of height h is observed from a point at distance d, the angle of elevation of the top satisfies tan θ = h/d. For two observation points on the same side, subtract the distances to get the unknown height.

Problem-solving tips

• Always convert the word problem into a right triangle first. Mark the vertical side as the height and the horizontal side as the distance.
• When two observation points lie on the same line as the foot of the tower, write one equation for each point. Both equations share the same height h, so equate them to find the unknown distance.
• If the observer's eye is above the ground (for example 1.5 m), add that eye height to the height found from the triangle.
• Keep the surds as they are (√3, 1/√3) until the last step, because the standard values then cancel neatly.
• In a problem with a shadow, the sun's angle of elevation is the angle between the ray of light and the ground. A pole of height h with shadow length s satisfies tan θ = h/s. A tower with a 60° sun elevation and a 30 m shadow has height 30 tan 60° = 30√3 m.
• A man 40 m from a building with a 45° elevation sees its top at 40 m above his eye. If his eye is 1.5 m high, the building is 41.5 m high.

Exam traps

• Measuring the angle of elevation from the vertical instead of from the horizontal.
• Using sin when the question gives the base and height (the correct ratio is tan).
• Swapping the values of sin 30° (1/2) and sin 60° (√3/2), or cos 30° and cos 60°.
• Thinking tan 90° has a value. It is not defined.
• Believing sin θ + cos θ = 1. Only sin²θ + cos²θ = 1 is an identity.
• Saying sec θ can be below 1 for an acute angle. Sec and cosec are always at least 1.
• Writing sin(90° − θ) = sin θ. The correct result is cos θ.
• Using the angle of depression as if it were measured from the ground. It is measured from the horizontal at the observer's eye.

One-liners

• 1. sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.
• 2. sin 30° = 1/2; cos 60° = 1/2; tan 45° = 1.
• 3. sin 60° = cos 30° = √3/2; tan 60° = √3; tan 30° = 1/√3.
• 4. sin²θ + cos²θ = 1.
• 5. 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
• 6. sin(90° − θ) = cos θ and tan(90° − θ) = cot θ.
• 7. sin θ and cos θ of an acute angle never exceed 1.
• 8. tan 0° = 0 and sin 0° = 0; cos 0° = 1; sin 90° = 1.
• 9. The angle of elevation is measured above the horizontal; the angle of depression below it.
• 10. The angle of depression from A to B equals the angle of elevation from B to A.
• 11. A 45° elevation gives height equal to the distance from the base.
• 12. For complementary elevations at distances a and b, height = √(ab).

Practice questions

1. The value of sin 30° is

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

   C. 1/2

   Standard value: sin 30° = 1/2.

2. The value of tan 45° is

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

   C. 1

   Opposite and adjacent are equal at 45°.

3. The value of cos 60° is

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

   B. 1/2

   Standard value: cos 60° = 1/2.

4. The value of tan 60° is

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

   D. √3

   Standard value: tan 60° = √3.

5. Which identity is true for every angle?

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

   B. sin²θ + cos²θ = 1

   This is the basic Pythagorean identity.

6. 1 + tan²θ equals

   1. cosec²θ
   2. cot²θ
   3. sin²θ
   4. sec²θ
   Answer

   D. sec²θ

   Dividing sin²θ + cos²θ = 1 by cos²θ gives 1 + tan²θ = sec²θ.

7. cosec²θ − cot²θ equals

   1. 0
   2. tan²θ
   3. sec²θ
   4. 1
   Answer

   D. 1

   From 1 + cot²θ = cosec²θ.

8. sin(90° − θ) equals

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

   B. cos θ

   Sine of the complement is the cosine.

9. If sin θ = 3/5 (θ acute), tan θ equals

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

   A. 3/4

   The adjacent side is 4, so tan θ = 3/4.

10. If tan θ = 5/12 (θ acute), sin θ equals

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

    B. 5/13

    The hypotenuse is 13, so sin θ = 5/13.

11. If sec θ = 5/3 (θ acute), tan θ equals

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

    A. 4/3

    tan²θ = 25/9 − 1 = 16/9.

12. If cos θ = 5/13 (θ acute), cosec θ equals

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

    D. 13/12

    sin θ = 12/13, so cosec θ = 13/12.

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

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

    B. 1

    1/4 + 3/4 = 1.

14. The value of cos²30° − sin²30° is

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

    B. 1/2

    3/4 − 1/4 = 1/2.

15. The value of tan 10° × tan 80° is

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

    D. 1

    tan 80° = cot 10°, so the product is 1.

16. The value of sin 25° / cos 65° is

    1. 1
    2. 0
    3. 1/2
    4. tan 40°
    Answer

    A. 1

    cos 65° = sin 25°.

17. If sin θ = cos θ for an acute angle θ, then θ equals

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

    C. 45°

    Sine and cosine are equal at 45°.

18. If tan θ = √3 for an acute angle, θ equals

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

    C. 60°

    tan 60° = √3.

19. The maximum possible value of sin θ is

    1. 1
    2. 1/2
    3. √2
    4. Not bounded
    Answer

    A. 1

    sin θ lies between −1 and 1, with the maximum 1 at 90°.

20. A 10 m ladder leans against a wall making 60° with the ground. The height reached on the wall is

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

    B. 5√3 m

    h = 10 sin 60° = 10 × √3/2 = 5√3 m.

21. The foot of a 10 m ladder making 60° with the ground is at what distance from the wall?

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

    D. 5 m

    d = 10 cos 60° = 5 m.

22. From a point 30 m from the foot of a tower, the angle of elevation of the top is 45°. The height of the tower is

    1. 30√3 m
    2. 30 m
    3. 15 m
    4. 60 m
    Answer

    B. 30 m

    h = 30 tan 45° = 30 m.

23. From a point 20√3 m from the foot of a tower, the angle of elevation of the top is 30°. The tower is how high?

    1. 60 m
    2. 20√3 m
    3. 20 m
    4. 10 m
    Answer

    C. 20 m

    h = 20√3 × tan 30° = 20√3 × 1/√3 = 20 m.

24. A kite string 100 m long makes an angle of 30° with the ground (string taut and straight). The height of the kite is

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

    A. 50 m

    h = 100 sin 30° = 50 m.

25. From the top of a lighthouse 75 m high, the angle of depression of a ship is 30°. The distance of the ship from the foot of the lighthouse is

    1. 150 m
    2. 25√3 m
    3. 75 m
    4. 75√3 m
    Answer

    D. 75√3 m

    d = 75 / tan 30° = 75√3 m.

26. A 10 m pole casts a shadow of 10√3 m. The angle of elevation of the sun is

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

    B. 30°

    tan θ = 10/(10√3) = 1/√3, so θ = 30°.

27. The angle of elevation of the top of a tower from two points on a line through its foot, 4 m and 9 m from the foot, are complementary. The height of the tower is

    1. 6 m
    2. 13 m
    3. 36 m
    4. 5 m
    Answer

    A. 6 m

    h = √(ab) = √36 = 6 m.

28. A tree is broken by a storm and its top touches the ground 10√3 m from the foot, making 30° with the ground. The original height of the tree is

    1. 10 m
    2. 20√3 m
    3. 30 m
    4. 20 m
    Answer

    C. 30 m

    The standing part is 10 m and the broken part is 10/sin 30° = 20 m. Total 30 m.

29. The angle of depression of an object from an observer equals the angle of elevation of the observer from the object because they are

    1. vertically opposite angles
    2. complementary angles
    3. co-interior angles
    4. alternate angles
    Answer

    D. alternate angles

    The two horizontals are parallel, so the angles are alternate angles.

30. The angle of elevation is measured from the

    1. ground at the object
    2. horizontal line through the eye
    3. line joining the object to the base
    4. vertical line through the eye
    Answer

    B. horizontal line through the eye

    Both elevation and depression are measured from the horizontal.

31. The value of 4 sin 30° cos 60° is

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

    D. 1

    4 × 1/2 × 1/2 = 1.

32. The value of (1 − cos²θ)/sin²θ is

    1. 1
    2. 0
    3. tan²θ
    4. cos²θ
    Answer

    A. 1

    1 − cos²θ = sin²θ.

33. Which of the following is NOT defined?

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

    C. tan 90°

    cos 90° = 0, so tan 90° = sin 90°/cos 90° is not defined.

34. Which of the statements is/are correct? 1. The values of sin θ and cos θ of an acute angle never exceed 1. 2. The value of sec θ of an acute angle can be less than 1.

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

    A. 1 only

    sec θ = 1/cos θ is at least 1 for an acute angle.

35. Which of the statements is/are correct? 1. cos(90° − θ) = sin θ. 2. cosec(90° − θ) = sec θ.

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

    C. Both 1 and 2

    Both are complementary angle relations.

36. Which of the statements is/are correct? 1. tan θ is always less than 1 for an acute angle. 2. sin θ = cos θ for θ = 30°.

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

    D. Neither 1 nor 2

    tan 60° = √3 > 1, and sin 30° = 1/2 while cos 30° = √3/2.

37. Which of the statements is/are correct? 1. sin θ + cos θ = 1 is an identity. 2. sin²θ + cos²θ = 1 is an identity.

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

    B. 2 only

    sin θ + cos θ equals √2 at 45°, not 1.

38. Which of the statements is/are correct? 1. The angle of elevation is measured above the horizontal. 2. The angle of depression is measured from the vertical.

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

    A. 1 only

    The angle of depression is also measured from the horizontal, below it.

39. Which of the statements is/are correct? 1. tan 30° = 1/√3. 2. sin 60° = cos 30°.

    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 values and a complementary relation.

40. Which of the statements is/are correct? 1. sin θ increases as θ goes from 0° to 90°. 2. cos θ increases as θ goes from 0° to 90°.

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

    A. 1 only

    cos θ decreases from 1 to 0 over that range.

41. Which of the statements is/are correct? 1. In a 45°–45°–90° triangle the sides are in the ratio 1 : 1 : √2. 2. In a 30°–60°–90° triangle the sides are in the ratio 1 : √3 : 2.

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

    C. Both 1 and 2

    Both are the standard ratios of these triangles.

42. Match the angles with their ratios: (a) sin 60°, (b) cos 45°, (c) tan 30°. The correct values in order are

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

    B. √3/2, 1/√2, 1/√3

    These are the standard values.

43. Match: (a) sec θ, (b) cosec θ, (c) cot θ. The correct definitions in order are

    1. 1/cos θ, 1/tan θ, sin θ/cos θ
    2. 1/sin θ, 1/cos θ, sin θ/cos θ
    3. cos θ, sin θ, tan θ
    4. 1/cos θ, 1/sin θ, cos θ/sin θ
    Answer

    D. 1/cos θ, 1/sin θ, cos θ/sin θ

    Sec, cosec and cot are reciprocals of cos, sin and tan.

44. A man standing 40 m from a building sees its top at an angle of elevation of 45°. His eye level is 1.5 m above the ground. The height of the building is

    1. 38.5 m
    2. 43 m
    3. 40 m
    4. 41.5 m
    Answer

    D. 41.5 m

    Height = 40 tan 45° + 1.5 = 41.5 m.

45. The shadow of a tower is 30 m long when the sun's elevation is 60°. The height of the tower is

    1. 60 m
    2. 30√3 m
    3. 30 m
    4. 10√3 m
    Answer

    B. 30√3 m

    h = 30 tan 60° = 30√3 m.