Put it all together. Ladders, shadows, ramps, kites and towers. Choose the right strategy, use exact values and verify your answers make sense.
Today's hook: A 5 m ladder leans against a wall at 60 degrees. Without calculating, estimate how high up the wall it reaches. Then check your estimate using exact values -- no calculator needed.
0/5QUESTS
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A 5 m ladder leans against a wall at $60°$. Without calculating, estimate how high up the wall it reaches. Then check your estimate using exact values.
Record your answer in your workbook.
1
The Big Idea
+5 XP to read
This lesson pulls together everything from the past six lessons. Exact values for 30°, 45° and 60° let you solve problems without a calculator. Choosing the right strategy -- Pythagoras, trig, or inverse trig -- is the difference between a quick solution and a dead end. Verifying reasonableness catches errors before they cost marks.
The three special angles (30°, 45°, 60°) come from two triangles: the isosceles right triangle (45°) and the half-equilateral triangle (30° and 60°). Memorising their exact values means you can solve many problems mentally, without reaching for a calculator.
30°, 45°, 60° -- sin, cos, tan. They appear constantly.
Choose before calculating
Decide Pythagoras vs trig before touching your calculator.
Sanity check every answer
A person's shadow should not be 150 m long.
2
What You'll Master
objectives
Know
The exact values for 30°, 45° and 60°
When to use Pythagoras versus trigonometry
Understand
How to choose the most efficient strategy for a given problem
That exact values avoid rounding errors
Can Do
Solve mixed word problems involving right-angled triangles
Use exact values where appropriate
Verify answers by checking reasonableness
3
Words You Need
vocabulary
Exact valueA value expressed using surds or fractions, not a rounded decimal.
Special triangleThe 30-60-90 and 45-45-90 triangles that produce exact trig values.
Reasonableness checkVerifying that your answer is sensible in the real-world context.
StrategyThe choice of mathematical tool for a given problem.
Inverse trigsin−1, cos−1, tan−1 -- used to find an angle from a ratio.
Pythagorean tripleA set of three integers that satisfy $a^2 + b^2 = c^2$, e.g. 3-4-5, 5-12-13.
4
Spot the Trap
heads-up
Wrong: Using Pythagoras when you need to find an angle. Pythagoras only finds sides.
Right: If you need an angle, you must use inverse trig. If you need a side, use Pythagoras or trig.
Wrong: Accepting an answer like "height = 150 m" when the object is a person standing next to a house.
Right: Always do a quick sanity check. Does the answer match the scale of the problem?
5
Parts of the Whole
+5 XP
Every mixed problem contains the same four pieces: a real-world scenario, a hidden right-angled triangle, the given information, and the question. Your job is to extract the triangle, label it, and choose the right tool.
Before calculating, draw the triangle, label the sides and angles, and identify what is given and what is wanted. Only then can you decide whether Pythagoras, trig, or inverse trig is the right move.
scenario to triangle to tool to answer
Draw before calculating
Every word problem hides a triangle. Draw it out.
Label everything
Mark known sides, known angles, and the unknown.
Check the scale
A shadow of 150 m from a person is not reasonable.
6
Exact Values
+5 XP
Memorising the exact values for 30 degrees, 45 degrees and 60 degrees saves time and eliminates calculator errors. These values come from two special triangles: the half-equilateral (30-60-90) and the isosceles right (45-45-90).
Notice the pattern: sin goes 1/2, root-2/2, root-3/2 as the angle increases. Cos goes backwards. Tan is sin divided by cos. If you know sin and cos, tan follows automatically.
Every right-angled triangle problem has exactly one best approach. The decision is based on what you know and what you want to find. Learn to spot it instantly.
Two sides, want third -- Pythagoras. One side + one angle, want another side -- trig ratio. Two sides, want angle -- inverse trig. All three sides, want angle -- any inverse trig ratio.
Two sides and no angle means Pythagoras is your only move.
Angle present = trig
Once an angle is involved, SOH CAH TOA unlocks everything.
Want angle = inverse
sin-1, cos-1, tan-1 turn ratios back into angles.
8
Check and Verify
+5 XP
A quick reasonableness check catches most errors before they cost marks. Ask three questions: is the angle between 0 and 90 degrees? Does the side length match the scale of the problem? Do the three angles sum to 180 degrees?
If a person's shadow calculates to 150 m, something is wrong. If a ladder angle comes out as 150 degrees, you have made an error. If the three angles of a triangle do not sum to 180 degrees, recheck your working.
$\theta_1 + \theta_2 + 90° = 180°$
Catch impossible angles
An acute angle over 90 degrees means an error.
Match the real world
A house roof should not be 1000 metres high.
Sum to 180 degrees
The two acute angles must sum to 90 degrees.
Watch Me Solve It · 3 examples
Watch Me Solve It · Using exact values
+15 XP per step
Q1
PROBLEM
An equilateral triangle has side length 10 cm. Find its height using exact values.
1
Draw and split the triangle
Split into two 30-60-90 right triangles
The perpendicular from a vertex bisects the opposite side in an equilateral triangle.
2
Label the right triangle
Hypotenuse = 10 cm, base = 5 cm, height = $h$, angle = 60°
sin 60° = root-3/2 is an exact value -- no calculator needed.
4
Approximate check
$5\sqrt{3} \approx 5 \times 1.732 \approx 8.66$ cm
The height should be less than the side length. 8.66 < 10 ✓
Nice work -- XP earned
Answer$h = 5\sqrt{3}$ cm $\approx 8.66$ cm
Watch Me Solve It · Square diagonal
+15 XP per step
Q2
PROBLEM
A square has diagonal length $8\sqrt{2}$ cm. Find its side length using exact values.
1
Identify the triangle
A diagonal splits a square into two isosceles right triangles
Each triangle has two equal sides (the square's sides) and a right angle.
2
Use the isosceles right triangle property
Diagonal = side $\times \sqrt{2}$
In a 45-45-90 triangle, hypotenuse = leg $\times \sqrt{2}$.
3
Solve for the side
$8\sqrt{2} = s \times \sqrt{2}$ → $s = 8$ cm
4
Verify
Side = 8, so diagonal = $8\sqrt{2}$ ✓
The root-2 factors cancel perfectly.
Nice work -- XP earned
Answer$8$ cm
Watch Me Solve It · Rope and pole
+15 XP per step
Q3
PROBLEM
A 12 m rope is tied from the top of a pole to a stake in the ground, making a 30° angle with the ground. Find the height of the pole using exact values.
1
Draw and label
Rope = hypotenuse = 12 m, angle with ground = 30°, height = $h$
The pole is vertical, the ground is horizontal, the rope is the hypotenuse.
2
Choose the ratio
Height is opposite the 30° angle, rope is hypotenuse → use sine
At 30°, the opposite side is exactly half the hypotenuse. 6 = 12/2 ✓
Nice work -- XP earned
Answer$h = 6$ m
9
Common Pitfalls
heads-up
Using Pythagoras to find an angle
Pythagoras' theorem only connects three sides. It cannot produce an angle. If the question asks for an angle, you must use inverse trig.
Fix: if the answer must be in degrees, inverse trig is required.
Forgetting exact values exist
Reaching for a calculator to find sin 30 degrees. The exact value is 1/2 -- instant, precise, and never wrong. Calculators can be in the wrong mode.
Fix: always check if the angle is 30, 45, or 60 before using a calculator.
Accepting absurd answers
Writing "height = 250 m" for a garden shed roof, or "angle = 89.9 degrees" for a shallow ramp. These answers reveal that something went wrong in the calculation.
Fix: after every answer, ask "does this make sense in the real world?"
Copy Into Your Books
Exact Values
sin 30 = 1/2, cos 30 = root3/2
sin 45 = cos 45 = 1/root2
sin 60 = root3/2, cos 60 = 1/2
tan 30 = 1/root3, tan 45 = 1, tan 60 = root3
Strategy Choice
2 sides, want 3rd -- Pythagoras
1 side + 1 angle -- trig ratio
2 sides, want angle -- inverse trig
Special Triangles
45-45-90: hyp = leg x root2
30-60-90: sides in ratio 1 : root3 : 2
Reasonableness
Acute angles: 0 to 90
Angles sum to 180
Match real-world scale
How are you completing this lesson?
Brain Trainer · 4 problems
D
Brain Trainer · Mixed
4 problems
Four problems using exact values and strategy choice. Work each one, then reveal the answer.
1 Write the exact value of $\sin 60°$.
From the 30-60-90 triangle, opposite = root3 and hypotenuse = 2.$\sin 60° = \frac{\sqrt{3}}{2}$
2 Write the exact value of $\cos 45°$.
From the 45-45-90 triangle, adjacent = 1 and hypotenuse = root2.$\cos 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
3 Write the exact value of $\tan 30°$.
From the 30-60-90 triangle, opposite = 1 and adjacent = root3.$\tan 30° = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
4 An equilateral triangle has side length 8 cm. Find its exact height.
Split into two 30-60-90 triangles. Hypotenuse = 8, so height = $8 \times \sin 60° = 8 \times \frac{\sqrt{3}}{2}$.$= 4\sqrt{3}$ cm
Complete in your workbook.
Multiple Choice · 5 questions
MC1
Exact value of sin 60 degrees
+10 XP
The exact value of $\sin 60°$ is:
Correct -- sin 60° = root-3/2 from the 30-60-90 triangle.
Not quite -- remember the pattern: sin increases from 1/2 to root-3/2.
Explanation: In a 30-60-90 triangle with hypotenuse 2, the side opposite 60° has length $\sqrt{3}$. Therefore $\sin 60° = \frac{\sqrt{3}}{2}$.
MC2
Smallest angle in a 6-8-10 triangle
+10 XP
A right-angled triangle has sides 6, 8, 10. The smallest angle is closest to:
Correct -- the smallest angle is opposite the shortest side (6). $\sin \theta = 6/10 = 0.6$ → $\theta \approx 36.9°$.
The smallest angle is opposite the shortest side. Use sin−1(6/10).
Explanation: The smallest angle is opposite the shortest side (6). $\sin \theta = \frac{6}{10} = 0.6$, so $\theta = \sin^{-1}(0.6) \approx 36.87° \approx 37°$.
MC3
Strategy for hypotenuse
+10 XP
Which strategy is best to find the hypotenuse when you know both legs?
Correct -- two sides and no angle means Pythagoras.
Trig ratios need an angle. Without one, use Pythagoras.
Explanation: Pythagoras' theorem ($a^2 + b^2 = c^2$) connects three sides. Trigonometric ratios need an angle to create a relationship between sides. With only side lengths given, Pythagoras is the only appropriate tool.
MC4
Isosceles right triangle hypotenuse
+10 XP
An isosceles right triangle has legs of length 5. Its hypotenuse is:
Correct -- hypotenuse = leg x root2 = $5\sqrt{2}$.
In a 45-45-90 triangle, the hypotenuse is always the leg times root2.
Explanation: $c^2 = 5^2 + 5^2 = 25 + 25 = 50$, so $c = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$. Option C simplifies to $5\sqrt{2}$ as well, but B is the standard form.
MC5
Ramp angle from rise and run
+10 XP
A ramp rises 1 m over a run of $\sqrt{3}$ m. The angle of elevation is:
Correct -- tan theta = 1/root3 → theta = 30°.
Calculate tan theta = opposite/adjacent = 1/root3.
Explanation: $\tan \theta = \frac{1}{\sqrt{3}}$. Since $\tan 30° = \frac{1}{\sqrt{3}}$, the angle is $30°$.
Short Answer · 3 questions
Q6
Shadow and sun angle
+15 XP
Q6
SHORT ANSWER
A person 1.8 m tall casts a shadow 3.1 m long. (a) Find the angle of elevation of the sun to the nearest degree. (b) If the shadow lengthens to 4.5 m, what is the new angle of elevation? (c) Explain why the angle decreases as the shadow lengthens.
(c) As the shadow lengthens, the adjacent side increases while the opposite side (height) stays constant. Since $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$, a larger denominator gives a smaller ratio, so the angle decreases.
Q7
Roof pitch and building code
+15 XP
Q7
SHORT ANSWER
A roof is built as an isosceles triangle with base 8 m and equal sides 5 m. (a) Find the height of the roof using Pythagoras. (b) Find the angle the roof makes with the horizontal. (c) Australian building codes require this angle to be between 15° and 22° for standard tiles. Does this roof meet the code?
Write your working in your book.
(a) Half base = 4 m. $h = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$ m.
(c) $37°$ is outside the 15-22 degree range, so it does not meet the code.
Q8
Zip line safety
+15 XP
Q8
SHORT ANSWER
A zip line runs from a platform 15 m high to a landing point on the ground 20 m horizontally away. (a) Find the length of the zip line. (b) Find the angle the zip line makes with the horizontal. (c) For safety, the angle must be between 30° and 45°. Does this zip line meet the safety requirement? (d) If the platform height is fixed at 15 m, what is the minimum horizontal distance needed for the zip line to be safe?
(c) $37°$ is between $30°$ and $45°$, so it meets the safety requirement.
(d) For max angle 45°: $\tan 45° = \frac{15}{d}$ → $d = \frac{15}{1} = 15$ m. For min angle 30°: $\tan 30° = \frac{15}{d}$ → $d = \frac{15}{1/\sqrt{3}} = 15\sqrt{3} \approx 26$ m. Minimum distance for safety = 15 m (at 45 degrees).
S
Stretch Challenge · Consistent measurements
+20 XP
S
STRETCH
A tower stands on level ground. From a point 50 m from the base, the angle of elevation to the top is 40°. From a point 80 m from the base, the angle of elevation is 25°. Are these measurements consistent with the same tower height? If not, which measurement is more likely wrong and why? Show all working.
Record in your book -- full marks require clear working.
From 50 m: $h = 50 \times \tan 40° \approx 50 \times 0.839 = 41.95$ m.
From 80 m: $h = 80 \times \tan 25° \approx 80 \times 0.466 = 37.28$ m.
The heights are not consistent ($41.95 \neq 37.28$).
The 25° measurement is more likely wrong. Measuring angles of elevation from farther away is harder because the angle is smaller and small errors in sighting create larger percentage errors. Also, 40° is a larger, easier-to-measure angle.
Interactive -- Mixed Trig Solver
explore
Use the interactive below to practise mixed right-angled triangle problems. Choose the strategy, enter your working, and check your answer. Try to beat your previous time.
Record two observations about how the interactive helps you choose strategies.
Consolidation Game -- Doodle Jump Quiz
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Jump your way to the top by answering questions on mixed right-angled triangle problems. The higher you climb, the harder the questions.
Lesson Complete
+10 XP
Mark this lesson as complete to earn your bonus XP.