← Unit 4 Trigonometry Review and Applications

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Trigonometry Review and Applications

⏱ 25 min📚 Year 9📈
Think First

From a point 50 m from a tower, the angle of elevation to the top is $35°$. The observer's eye is 1.5 m above ground. Find the tower height.

💡 Revision: Ensure you can use SOH CAH TOA, inverse trig functions, and solve elevation/depression problems.

Learning Intentions

Know

  • SOH CAH TOA
  • Inverse trig
  • Elevation and depression
  • Problem solving

Understand

  • How to choose the correct trig ratio
  • When to add observer height

Can Do

  • Solve any right-angled triangle problem
  • Apply trigonometry to real-world scenarios
  • Check answers for reasonableness
SOH CAH TOAInverseElevationDepression
Learn Phase
1

Choosing the Right Ratio

Match given and wanted

To choose a trig ratio:

  1. Label the sides: opposite, adjacent, hypotenuse
  2. Identify what you know and what you want
  3. Choose the ratio connecting them
KnowWantUse
Angle, oppositeHypotenuseSine
Angle, adjacentHypotenuseCosine
Angle, oppositeAdjacentTangent
2

Elevation and Depression Checklist

Systematic problem solving

For elevation/depression problems:

  1. Draw a clear diagram
  2. Draw horizontal lines through observer and object
  3. Identify the angle (elevation or depression)
  4. Form a right triangle with the horizontal distance
  5. Use appropriate trig ratio
  6. Add observer height if needed for total height

Remember: angle of elevation = angle of depression (alternate angles).

3

Checking Answers

Reasonableness and verification

Always verify:

  • Is the answer physically reasonable? (e.g., height should be positive)
  • Does Pythagoras hold for calculated sides?
  • Can you verify with a different method?
  • Does the angle sum to 180° in the triangle?

Example: If you calculate a side longer than the hypotenuse, recheck — the hypotenuse is always longest.

✓

Check Understanding

Try it yourself

A ship is 80 m from a lighthouse. The angle of elevation to the top is $25°$. The observer is 2 m above sea level. Find the lighthouse height.

Worked Example

Trig Applications

1

A ramp is 10 m long and rises 2 m. Find the angle of inclination.

$sin heta = 2/10 = 0.2$

$ heta = sin^{-1}(0.2) approx 11.5°$

2

From a cliff 45 m high, the angle of depression to a boat is $18°$. How far is the boat?

$ an 18° = 45/d$

$d = 45/ an 18° approx 138.4$ m

3

A kite string is 60 m at $50°$ to horizontal. How high is the kite? (Assume string starts at ground)

$sin 50° = h/60$

$h = 60 imes sin 50° approx 45.96$ m

Common Misconceptions

✕

Use the wrong trig ratio. Always label sides first, then match to SOH CAH TOA.

✕

Forget to add observer height. If the observer is above ground, add their eye height to calculated vertical distances.

✕

Confuse angle of elevation with the angle inside the triangle. The elevation angle is measured from the horizontal, which is one angle of the right triangle.

Your Turn

Practice — Trig Review

Work through each question in your book or digitally. Answers are in the Questions phase.

1A 15 m ladder leans against a wall at $70°$. How high does it reach?
2From a window 12 m up, the angle of depression to a car is $30°$. How far is the car from the building?
3A triangle has angles $40°$ and $50°$, and the side opposite $40°$ is 8 cm. Find the hypotenuse.
Real-World Anchor

Surveying and Construction

Australian surveyors use trigonometry for land measurement, road design, and construction. The angle of elevation to a distant landmark, combined with known distance, allows calculation of height without direct measurement. Bridge and building designs rely on precise trigonometric calculations.

📓 Copy Into Your Books

▼

SOH CAH TOA

  • Sin = opp/hyp
  • Cos = adj/hyp
  • Tan = opp/adj

Elevation/Depression

  • Draw diagram
  • Horizontal lines
  • Angle from horizontal
  • Add observer height

Check

  • Reasonable answer?
  • Pythagoras check
  • Try different method
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
To find angle given opposite and hypotenuse, use:
Angle of elevation is measured from:
A 10 m ladder at $60°$ reaches:
From 30 m cliff, depression $20°$. Distance to object:
If $sin heta = 0.6$, then $cos heta = $
Kite string 50 m at $40°$. Height:
Elevation angle = depression angle because:
Observer 1.5 m tall, 40 m from tree, elevation $35°$. Tree height:
1A boat is 100 m from a lighthouse base. The angle of elevation to the top is $15°$. The observer is on a 5 m cliff. Find the lighthouse height.
2A ramp rises 1.5 m over 12 m horizontal. Find the angle of inclination.
3Explain how to verify a trigonometry calculation using a different ratio.

Comprehensive Answers

1Boat 100 m, elevation $15°$, observer 5 m.
$h = 100 imes an 15° + 5 approx 26.8 + 5 = 31.8$ m.
2Ramp 1.5 m over 12 m.
$ an heta = 1.5/12 = 0.125$, $ heta approx 7.1°$.
3Verify with different ratio.
Calculate another side using a different ratio, then check with Pythagoras.
MC 1Find angle given opp and hyp.
Inverse sine. Answer: C
MC 2Elevation measured from.
Horizontal. Answer: B
MC 310 m ladder at $60°$.
$10 imes sin 60° approx 8.66$ m. Answer: B
MC 430 m cliff, $20°$ depression.
$30/ an 20° approx 82.4$ m. Answer: C
MC 5$sin = 0.6$, find $cos$.
$cos = sqrt{1-0.36} = 0.8$. Answer: C
MC 6Kite 50 m at $40°$.
$50 imes sin 40° approx 32.1$ m. Answer: A
MC 7Elevation = depression.
Alternate angles. Answer: A
MC 8Tree height with observer.
$40 imes an 35° + 1.5 approx 28.0 + 1.5 = 29.5$ m. Answer: B
SA 1Lighthouse height.
$100 imes an 15° + 5 approx 31.8$ m.
SA 2Ramp angle.
$ heta approx 7.1°$.
SA 3Verify with different ratio.
Use alternative ratio, check with Pythagoras.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.