NCEA Level 1Trigonometric ratios in 2D and 3DAS 91947

Trigonometry: 7 exam questions solved step by step

What this lesson covers

  • In similar right-angled triangles the ratios of sides are fixed for a given angle, which is what sine, cosine and tangent name.
  • Choosing the ratio depends on which two sides are known or wanted relative to the angle.
Key idea

7 questions, one method

  • 1. Label opp, adj, hyp from the angle.
  • 2. SOH CAH TOA: pick the ratio.
  • 3. Solve, then round at the end.
  • Calculator in degrees.
Your turn

Question 1 · Achieved

  • Hypotenuse 15 cm, angle 40°.
  • Find x, the side opposite 40°.
Question 1 · Achieved
Worked example

Q1: x opposite 40°, hyp 15 cm

  • Opp and hyp: SOH.
  • x on top: multiply.
  1. opp = x, hyp = 15adj not used
  2. sin 40° = x ÷ 15SOH
  3. x = 15 × sin 40°× 15 both sides
  4. x = 9.6418…calculator
  5. x = 9.64 cm (3 s.f.)
Your turn

Question 2 · Achieved

  • Hypotenuse 14 cm.
  • The side next to θ is 9 cm.
  • Find θ to 1 decimal place.
Question 2 · Achieved
Worked example

Q2: find θ

  • Adj and hyp: CAH.
  • Angle: use cos⁻¹.
  1. adj = 9, hyp = 14opp not used
  2. cos θ = 9 ÷ 14CAH
  3. θ = cos⁻¹(9 ÷ 14)inverse cosine
  4. θ = 49.9948…°calculator
  5. θ = 50.0° (1 d.p.)
Your turn

Question 3 · Merit

  • Angle 32°, opposite side 6 cm.
  • Find x, the side next to 32°.
  • Careful: x ends up on the bottom.
Question 3 · Merit
Worked example

Q3: x adjacent to 32°

  • Opp and adj: TOA.
  • x on the bottom: divide.
  1. opp = 6, adj = xhyp not used
  2. tan 32° = 6 ÷ xTOA
  3. x = 6 × tan 32° = 3.75x is on the bottom
  4. x × tan 32° = 6× x both sides
  5. x = 6 ÷ tan 32°÷ tan 32° both sides
  6. x = 9.6020…calculator
  7. x = 9.60 cm (3 s.f.)
Your turn

Question 4 · Merit, in context

  • You stand 120 m from a tower.
  • Angle of elevation to the top: 70°.
  • How tall is the tower?
Question 4 · Merit, in context
Worked example

Q4: elevation 70°, 120 m away

  • Elevation: up from flat.
  • Opp and adj: TOA.
  1. opp = h, adj = 120hyp not used
  2. tan 70° = h ÷ 120TOA
  3. h = 120 × tan 70°× 120 both sides
  4. h = 329.697… mcalculator
  5. h = 330 m (3 s.f.)
  6. The tower is about 330 m tall.
Your turn

Question 5 · Merit, in context

  • A 6 m ladder leans on a wall.
  • Its foot is 1.5 m from the wall.
  • Safe if its angle with the ground
  • is between 70° and 80°. Is it safe?
Question 5 · Merit, in context
Worked example

Q5: is the ladder safe?

  • Adj and hyp: CAH.
  • Compare, answer in words.
  1. adj = 1.5, hyp = 6opp not used
  2. cos θ = 1.5 ÷ 6 = 0.25CAH
  3. θ = cos⁻¹(0.25)inverse cosine
  4. θ = 75.522…°calculator
  5. θ = 75.5° (1 d.p.)
  6. 70° < 75.5° < 80°
  7. 75.5°, so the ladder is safe
Your turn

Question 6 · Excellence

  • From the top of a 40 m cliff:
  • angle of depression to a boat 30°,
  • to a boat further out 20°.
  • How far apart are the boats?
Question 6 · Excellence
Worked example

Q6: the near boat

  • Alternate angles first.
  • Keep every digit.
  1. angle at near boat = 30°alternate angles
  2. angle at far boat = 20°alternate angles
  3. tan 30° = 40 ÷ aTOA
  4. a = 40 ÷ tan 30°a on the bottom
  5. a = 69.282… mdo not round yet
Worked example

Q6: the far boat, then the gap

  • Same method for b.
  • Subtract, then round.
  1. a = 69.282… m
  2. tan 20° = 40 ÷ bTOA
  3. b = 40 ÷ tan 20°b on the bottom
  4. b = 109.899… mcalculator
  5. d = b − afar take away near
  6. = 109.899… − 69.282…
  7. d = 40.617… m
  8. 110 − 69 = 41rounded too early
  9. The boats are 40.6 m apart.
Your turn

Question 7 · Excellence

  • A cuboid room: 5 m by 4 m, 3 m high.
  • A string joins a floor corner
  • to the opposite ceiling corner.
  • Find its angle with the floor.
Question 7 · Excellence
Worked example

Q7: the floor diagonal

  • Not the edge: the diagonal.
  • Pythagoras first.
  1. tan θ = 3 ÷ 5edge, not diagonal
  2. diagonal² = 5² + 4²Pythagoras on floor
  3. = 25 + 16 = 41
  4. diagonal = √41 = 6.403… mkeep it as √41
Worked example

Q7: the angle with the floor

  • Upright triangle on it.
  • Opp and adj: TOA.
  1. diagonal = √41 m
  2. opp = 3, adj = √41hyp is the string
  3. tan θ = 3 ÷ √41TOA
  4. θ = tan⁻¹(3 ÷ √41)inverse tangent
  5. θ = 25.104…°calculator
  6. θ = 25.1° (1 d.p.)
  7. String to floor: 25.1°
Summary

The method that solved all seven

  • Label, pick the ratio, solve, and round last.
StepHow
Label sidesopp, adj, hyp from the angle
Pick ratioSOH CAH TOA, from two sides
Side on topmultiply: 15 × sin 40°
Side belowdivide: 6 ÷ tan 32°
Find anglesin⁻¹, cos⁻¹ or tan⁻¹
Depressionalternate angles: same at boat
In a cuboidPythagoras on the floor first
Round lastkeep every digit until the end

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