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State Board · Class 10 · Mathematics Part 2

Lesson 6: Trigonometry

3 concepts · 24 practice questions with answers · Free to read

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Trigonometric ratios

From daily life

When a ladder is leaned on a wall, the more it is tilted the higher it reaches. The link between the tilt and the ratios of the sides is trigonometry.

Simple definition

In a right triangle, for the angle θ there is the opposite side, the adjacent side and the hypotenuse. sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent. Their reciprocals: cosec θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ.

θ435sin θ = 3/5cos θ = 4/5tan θ = 3/4

In the figure: Sides 3, 4, 5; opposite θ is 3, adjacent is 4, hypotenuse is 5. sin θ = 3/5, cos θ = 4/5, tan θ = 3/4.

Rule 1: Find sin, cos and tan — Identify the angle θ and pick the opposite, the adjacent and the hypotenuse. Use Pythagoras if a side is missing.

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Trigonometric ratios

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  • Easy: Opposite 3, adjacent 4, hypotenuse 5 ⇒ sin θ = 3/5, cos θ = 4/5, tan θ = 3/4.
  • Medium: Opposite 5, adjacent 12, hypotenuse 13 ⇒ sin θ = 5/13, cos θ = 12/13, tan θ = 5/12.
  • Exam-type: sin θ = 8/17 ⇒ opposite 8, hypotenuse 17, adjacent = √(289 − 64) = 15 ⇒ cos θ = 15/17, tan θ = 8/15.

Rule 2: The reciprocal ratios — cosec, sec and cot are the reciprocals (1 ÷) of sin, cos and tan; just flip the fraction.

  • Easy: sin θ = 3/5, cos θ = 4/5, tan θ = 3/4 ⇒ cosec θ = 5/3, sec θ = 5/4, cot θ = 4/3.
  • Medium: sin θ = 5/13, cos θ = 12/13, tan θ = 5/12 ⇒ cosec θ = 13/5, sec θ = 13/12, cot θ = 12/5.
  • Exam-type: tan θ = 7/24 ⇒ hypotenuse = √(49 + 576) = 25 ⇒ sin θ = 7/25, cos θ = 24/25, cosec θ = 25/7, sec θ = 25/24.

Rule 3: Values for 30°, 45° and 60° — sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2. The cos values are in the reverse order. tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

  • Easy: sin 30° + cos 60° = 1/2 + 1/2 = 1.
  • Medium: tan 60° × tan 30° = √3 × 1/√3 = 1.
  • Exam-type: 3 tan² 30° + 2 sin² 60° = 3 × 1/3 + 2 × 3/4 = 1 + 3/2 = 5/2.
Remember

When θ changes, the opposite and adjacent sides change; the hypotenuse is always the longest side, opposite the right angle.

Common mistake

sin θ does not mean sin × θ; it is one ratio. Also sin 60° ≠ 2 × sin 30°.

Shortcut

Memory clue: "SOH-CAH-TOA" — sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. For 30°, 45°, 60° sin is √1/2, √2/2, √3/2; cos is the reverse.

Spot the difference

Pointsin 30° 45° 60°cos 30° 45° 60°
30°1/2√3/2
45°1/√21/√2
60°√3/21/2
0 to 1IncreasesDecreases

Practice questions

Easy
  1. 1Opposite 6, hypotenuse 10; what is sin θ?
  2. 2What is sin 30°?
  3. 3What is tan 45°?
Medium
  1. 4sin θ = 3/5; find cos θ and tan θ.
  2. 5cos θ = 5/13; find sin θ and cot θ.
  3. 6Evaluate: sin 60° × cos 30° − cos 60° × sin 30°.
Challenging
  1. 7Evaluate: 4 sin² 30° + tan² 60° − 2 cos² 45°.
  2. 8In a right triangle tan θ = 20/21. Find sin θ and sec θ.
Show answers (try them yourself first)
  1. 16/10 = 3/5.
  2. 21/2.
  3. 31.
  4. 4Adjacent = 4; cos θ = 4/5, tan θ = 3/4.
  5. 5Opposite = 12; sin θ = 12/13, cot θ = 5/12.
  6. 6(√3/2)(√3/2) − (1/2)(1/2) = 3/4 − 1/4 = 1/2.
  7. 74 × 1/4 + 3 − 2 × 1/2 = 1 + 3 − 1 = 3.
  8. 8Hypotenuse = √(400 + 441) = 29; sin θ = 20/29, sec θ = 29/21.

Trigonometric identities

From daily life

If two numbers add up to 10 and one is known, the other is known at once. In the same way, if sin θ is known, cos θ follows at once from a formula.

Simple definition

Equations that are true for every angle θ are called identities: sin² θ + cos² θ = 1; 1 + tan² θ = sec² θ; 1 + cot² θ = cosec² θ.

cos θsin θ1sin² θ + cos² θ = 1

In the figure: In a right triangle with hypotenuse 1 the sides are sin θ and cos θ; Pythagoras ⇒ sin² θ + cos² θ = 1.

Rule 1: sin² θ + cos² θ = 1 — If one is known, the square of the other = 1 − (square of the known one); then take the square root.

  • Easy: sin θ = 3/5 ⇒ cos² θ = 1 − 9/25 = 16/25 ⇒ cos θ = 4/5.
  • Medium: sin θ = 5/13 ⇒ cos² θ = 1 − 25/169 = 144/169 ⇒ cos θ = 12/13.
  • Exam-type: cos θ = 7/25 ⇒ sin² θ = 1 − 49/625 = 576/625 ⇒ sin θ = 24/25.

Rule 2: 1 + tan² θ = sec² θ — If tan θ is known, sec² θ = 1 + tan² θ; if cot is known, 1 + cot² θ = cosec² θ.

  • Easy: tan θ = 3/4 ⇒ sec² θ = 1 + 9/16 = 25/16 ⇒ sec θ = 5/4.
  • Medium: tan θ = 5/12 ⇒ sec² θ = 1 + 25/144 = 169/144 ⇒ sec θ = 13/12.
  • Exam-type: cot θ = 4/3 ⇒ cosec² θ = 1 + 16/9 = 25/9 ⇒ cosec θ = 5/3.

Rule 3: Simplify using identities — Spot the pairs: sin² + cos² = 1; sec² − tan² = 1; (1 − sin)(1 + sin) = 1 − sin² = cos². Writing everything in sin and cos is often easy.

  • Easy: 5 sin² θ + 5 cos² θ = 5 (sin² θ + cos² θ) = 5 × 1 = 5.
  • Medium: (1 − sin θ)(1 + sin θ) = 1 − sin² θ = cos² θ.
  • Exam-type: (1 + tan² θ) cos² θ = sec² θ × cos² θ = (1 ÷ cos² θ) × cos² θ = 1.
Remember

An identity is true for all angles; to check, try 30° or 45°.

Common mistake

sin² θ means (sin θ)², not sin (θ²). Also (sin θ + cos θ)² ≠ sin² θ + cos² θ.

Shortcut

Memory clue: "sin-cos 1, tan-sec 1, cot-cosec 1" — in each the 1 is added to the square of one ratio. Take the positive root (acute angle).

Spot the difference

Pointsin² + cos²1 + tan²
Equals1sec² θ
Pairsin and costan and sec
UseFind cosFind sec
Other formcos² = 1 − sin²tan² = sec² − 1

Practice questions

Easy
  1. 1What is sin² θ + cos² θ?
  2. 2sin θ = 4/5; what is cos θ?
  3. 3What is sec² θ − tan² θ?
Medium
  1. 4tan θ = 8/15; find sec θ.
  2. 5Simplify: (1 − cos θ)(1 + cos θ).
  3. 6Simplify: sin² θ + cos² θ + tan² θ.
Challenging
  1. 7Prove: (1 + tan² θ) cos² θ = 1.
  2. 8cos θ = 20/29; find tan θ and cosec θ.
Show answers (try them yourself first)
  1. 11.
  2. 21 − 16/25 = 9/25 ⇒ cos θ = 3/5.
  3. 31.
  4. 41 + 64/225 = 289/225 ⇒ sec θ = 17/15.
  5. 51 − cos² θ = sin² θ.
  6. 61 + tan² θ = sec² θ.
  7. 7LHS = sec² θ × cos² θ = (1 ÷ cos² θ) × cos² θ = 1.
  8. 8sin² θ = 1 − 400/841 = 441/841 ⇒ sin θ = 21/29; tan θ = 21/20, cosec θ = 29/21.

Heights and distances

From daily life

We do not climb a tree to measure its height. We stand a little away and look at the top, and use the angle and the distance to find the height.

Simple definition

The angle when we look up from the level of the eye is the angle of elevation, and when we look down it is the angle of depression. Both are measured from the horizontal line. tan θ = height ÷ distance.

θdhtan θ = h ÷ d

In the figure: The distance between the observer and the foot of the tower is d, the height is h, the angle of elevation is θ; tan θ = h ÷ d.

Rule 1: Find the height — h = d × tan θ. Take √3 = 1.73.

Live figure

Heights and distances

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  • Easy: Distance 10 m, θ = 45° ⇒ h = 10 × 1 = 10 m.
  • Medium: Distance 20 m, θ = 30° ⇒ h = 20 × 1/√3 = 20 ÷ 1.73 ≈ 11.56 m.
  • Exam-type: Distance 30 m, θ = 60° ⇒ h = 30 × √3 = 30 × 1.73 = 51.9 m.

Rule 2: Find the distance — d = h ÷ tan θ. If the height and the angle are known, the distance can be found.

  • Easy: h = 15 m, θ = 45° ⇒ d = 15 ÷ 1 = 15 m.
  • Medium: h = 12 m, θ = 60° ⇒ d = 12 ÷ 1.73 ≈ 6.94 m.
  • Exam-type: h = 10 m, θ = 30° ⇒ d = 10 × 1.73 = 17.3 m.

Rule 3: Angle of depression and the ladder — Angle of depression = the same angle of elevation (alternate angles). For a ladder: height = ladder × sin θ, foot distance = ladder × cos θ.

  • Easy: From a 30 m tall building the angle of depression of a car is 45° ⇒ distance = 30 ÷ 1 = 30 m.
  • Medium: From a 20 m high hill the angle of depression of a boat is 60° ⇒ distance = 20 ÷ 1.73 ≈ 11.56 m.
  • Exam-type: A 10 m ladder makes 30° with the ground ⇒ height = 10 × 1/2 = 5 m; foot distance = 10 × 0.865 = 8.65 m (√3/2 = 0.865).
Remember

The angle is always measured from the horizontal, not from the vertical. Draw the figure first.

Common mistake

If the height of the observer is given, do not forget to add or subtract it.

Shortcut

Memory clue: 45° ⇒ height = distance; 30° ⇒ long distance; 60° ⇒ tall height (√3 times). "Thirty is long, sixty is tall".

Spot the difference

PointAngle of elevationAngle of depression
DirectionUpDown
FromObserver's eyeA high place
Measured fromHorizontal lineHorizontal line
RelationEquals the depressionEquals the elevation

Practice questions

Easy
  1. 1Distance 12 m, θ = 45°; what is the height?
  2. 2tan θ = height ÷ ? (fill the blank)
  3. 3If the angle of depression is 30°, what is the angle of elevation?
Medium
  1. 4Distance 15 m, θ = 60°; find the height (√3 = 1.73).
  2. 5Tower height 30 m; elevation 30°; find the distance (√3 = 1.73).
  3. 6From a 40 m high lighthouse the angle of depression of a boat is 45°; how far is the boat?
Challenging
  1. 7A 14 m ladder leans on a wall at 60° with the ground. How high does it reach? (√3 = 1.73)
  2. 8An observer's eyes are 2 m above the ground. From 20 m from a tower the angle of elevation of the top is 45°; find the height of the tower.
Show answers (try them yourself first)
  1. 112 × 1 = 12 m.
  2. 2distance.
  3. 330° (alternate angles).
  4. 415 × 1.73 = 25.95 m.
  5. 530 × 1.73 = 51.9 m.
  6. 640 ÷ 1 = 40 m.
  7. 7Height = 14 × √3/2 = 7 × 1.73 = 12.11 m.
  8. 8Height above the eyes = 20 × 1 = 20 m; total = 20 + 2 = 22 m.