The triangle files · Goblin Detective Agency

Step 1 · Name your suspects (the three sides)

Before any trig, you label the three sides of the right-angled triangle. Two of the names depend on which angle you are standing at, so always find your angle first.

Stand at the marked angle θ. Everything is named from where YOU are standing.

Hypotenuse = the longest side, always opposite the right angle. It never changes, no matter which angle you stand at.
Opposite = the side across the triangle from your angle. You could throw a paper plane at it.
Adjacent = the side touching your angle that is NOT the hypotenuse. Adjacent means "next to".

🌶️ The trap Opposite and adjacent swap places if you move to the other angle. The hypotenuse is the only side with a permanent name. Label the sides fresh for every question, from the angle the question uses.

Step 2 · Meet the three informants

Each one is a ratio: one side divided by another. For a given angle, that ratio is ALWAYS the same, no matter how big the triangle is. That is the whole trick. Your calculator has all three memorised for every angle.

Informant	What they tell you	Memory hook
sin θ	oppositehypotenuse	SOH
cos θ	adjacenthypotenuse	CAH
tan θ	oppositeadjacent	TOA

🧌

Goblin says: SOH CAH TOA. Say it out loud three times. Some detectives remember it as "Some Old Hags Can't Always Hide Their Old Age", pick whichever sticks.

Picking the right informant

Cross out the side you do not know and do not care about. The two sides left in the question tell you who to call:

Got (or want) opposite and hypotenuse? Call sin.
Got (or want) adjacent and hypotenuse? Call cos.
Got (or want) opposite and adjacent? Call tan.

🌶️ Calculator check, do this NOW Your calculator must be in DEGREES mode (a little D or DEG on screen). Test it: sin 30 should give exactly 0.5. If it gives −0.988, you are in radians and every answer will be wrong. Fix the mode before anything else.

CASE T1 🏷️ Line-up Case Spice

A right-angled triangle has its right angle at the bottom right. The marked angle θ is at the bottom left. The three sides are: the slanted side (14 cm), the bottom side (11 cm) and the vertical side (8.7 cm). Name each side (hypotenuse, opposite, adjacent), then say which informant connects θ, 11 cm and 14 cm.

No calculator needed. This case is pure labelling, and labelling is where most triangle crimes happen.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Start with the easy one: the hypotenuse is opposite the right angle, and it is always the longest. Then stand at θ and point across the triangle for the opposite.

✅ Show the full solution

Hypotenuse = the slanted 14 cm side (opposite the right angle, and the longest).
Opposite = the vertical 8.7 cm side (across the triangle from θ).
Adjacent = the bottom 11 cm side (touching θ, not the hypotenuse).
11 cm is the adjacent and 14 cm is the hypotenuse. Adjacent and hypotenuse together mean the informant is cos (CAH).

👃 Sniff test The hypotenuse is the longest side, and 14 is the biggest of the three numbers. The labels hold up.

Case closed ✓

📓 Case Notes

Which side name never changes, and why?
If θ moved to the top corner instead, which two labels would swap?

CASE T2 🪜 Find-a-Side Case Spice

A goblin leans a 4 m ladder against the agency wall. The ladder makes an angle of 65° with the ground. How far up the wall does the ladder reach? Round to two decimal places.

Before you solve: sketch it. Wall vertical, ground flat, ladder slanted. Mark the 65° where the ladder meets the ground.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Stand at the 65° angle. The ladder is the slanted side, so it is the hypotenuse (4 m). The height up the wall is across the triangle from you, so it is the opposite, and it is what you want. Opposite and hypotenuse: which informant is that?

🔍 Show me a smaller case

Warm up: a 10 m ladder at 30°. How high? Opposite and hypotenuse means sin: height = 10 × sin 30° = 10 × 0.5 = 5 m.

Now do your real case exactly the same way with 4 and 65°.

✅ Show the full solution

Decode. Angle = 65°. Hypotenuse = 4 m (the ladder). Want = opposite (height up the wall), call it h. Opposite + hypotenuse = sin (SOH).

sin 65° = h ÷ 4

The unknown is on top of the fraction, so multiply both sides by 4: h = 4 × sin 65°.
Calculator (degrees mode!): sin 65° ≈ 0.9063.
h = 4 × 0.9063... = 3.6252...
Round: h ≈ 3.63 m.

🌶️ The trap Round at the END, not in the middle. If you round sin 65° to 0.9 early, you get 3.6 and lose marks. Keep everything in the calculator until the final line.

👃 Sniff test The height must be SHORTER than the ladder itself, and 3.63 m is less than 4 m. A steep 65° ladder should reach most of its length up the wall, and 3.63 out of 4 is most of it. It makes sense.

Case closed ✓

📓 Case Notes

How did you know it was sin and not cos?
What would the cos version of this question ask for instead? (Hint: along the ground.)

CASE T3 ♿ Upside-down Case Spice

The agency is building a ramp up to its front door, which is 1.2 m above the ground. Safety rules say the ramp must rise at an angle of 25°. How long does the ramp itself need to be? Round to two decimal places.

Careful: this time the unknown is the slanted side. That changes the last step, which is why this case gets its own file.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Stand at the 25°. The 1.2 m rise is across from you, so it is the opposite. The ramp is the slanted side, so it is the hypotenuse, and it is what you want. Opposite and hypotenuse is sin again, but this time the unknown is on the BOTTOM of the fraction.

🔍 Show me a smaller case

Warm up: a rise of 3 m at 30°. How long is the slope? sin 30° = 3 ÷ slope, so slope = 3 ÷ sin 30° = 3 ÷ 0.5 = 6 m.

See the flip? When the unknown is on the bottom, you end up DIVIDING by the trig value. Now do your real case the same way.

✅ Show the full solution

Decode. Angle = 25°. Opposite = 1.2 m (the rise). Want = hypotenuse (the ramp), call it r. Opposite + hypotenuse = sin (SOH).

sin 25° = 1.2 ÷ r

The unknown is on the bottom, so swap it with the trig value: r = 1.2 ÷ sin 25°.
Calculator: sin 25° ≈ 0.4226.
r = 1.2 ÷ 0.4226... = 2.8395...
Round: r ≈ 2.84 m.

🌶️ The trap The classic mistake is doing 1.2 × sin 25° = 0.51 m out of habit. But a ramp shorter than the height it climbs is impossible. When the unknown is the bottom of the fraction, you divide. Multiply when the unknown is on top, divide when it is on the bottom.

👃 Sniff test The ramp is the hypotenuse, so it MUST be longer than the 1.2 m rise, and 2.84 m is. A shallow 25° angle should need a longish ramp for a small rise. It makes sense.

Case closed ✓

📓 Case Notes

What is the tell that you divide instead of multiply?
Why is a hypotenuse answer smaller than a given side always wrong?

CASE T4 🌳 Mystery-Angle Case Spice

A tree outside the agency is 9 m tall and casts a shadow 12 m long on flat ground. At what angle would you have to look up, from the tip of the shadow, to see the top of the tree? Round to the nearest degree.

A spicy one: this time you KNOW two sides and the angle itself is the mystery. That needs a new calculator move.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Stand at the tip of the shadow. The tree (9 m) is across from you, the shadow (12 m) runs along the ground next to you. Opposite and adjacent: that is tan territory. But tan θ = 0.75 does not finish the job. To pull the angle OUT, you need the un-tan button: tan⁻¹ (usually SHIFT then tan).

🔍 Show me a smaller case

Warm up: a 1 m stick casts a 1 m shadow. What is the angle? tan θ = 1 ÷ 1 = 1, so θ = tan⁻¹(1) = 45°. That feels right: equal sides, halfway-up angle.

Now do your real case with 9 and 12.

✅ Show the full solution

Decode. Opposite = 9 m (tree). Adjacent = 12 m (shadow). Want = the angle θ. Opposite + adjacent = tan (TOA).

tan θ = 9 ÷ 12 = 0.75

The angle is locked inside the tan. Unlock it with the inverse: θ = tan⁻¹(0.75).
Calculator: SHIFT, tan, 0.75. Result: 36.869...°
Round: θ ≈ 37°.

🌶️ The trap tan⁻¹ does NOT mean 1 ÷ tan. It is a completely different button: the "undo" for tan, the same way the Backwards Cases undo operations in reverse. Also: do 9 ÷ 12, not 12 ÷ 9. The opposite goes on top.

👃 Sniff test The tree (9) is shorter than the shadow (12), so the angle should be LESS than 45°, and 37° is. Check it forwards: tan 37° ≈ 0.754, close to 0.75. It makes sense.

Case closed ✓

📓 Case Notes

How do you know a question is a Mystery-Angle Case before you start?
What does tan⁻¹ undo, in your own words?
Why did you expect an answer under 45° before you pressed anything?

The interrogation manual: one rule, three informants

CASE T2, T3 and T4 each hid one move inside their reveal drawer. Here is the whole manual on one page, and it works identically whether the informant is sin, cos or tan.

Where is the mystery?	The move	Example
Unknown on TOP of the fraction	Multiply	x = 12 × cos 40°
Unknown on the BOTTOM	Divide (swap it with the trig value)	x = 5 ÷ tan 28°
The ANGLE itself is locked inside	Inverse (SHIFT, then the button)	θ = sin⁻¹(0.62)

🧌

Goblin says: the informant is chosen by the two SIDES in the question. The move is chosen by where the MYSTERY sits. Two decisions, always in that order.

🌶️ The trap sin⁻¹ never means 1 ÷ sin, and the same goes for cos⁻¹ and tan⁻¹. They are undo buttons, exactly as CASE T4 warned you. If your working ever says 1 ÷ sin, close the file and start again.

CASE T5 🛬 Find-a-Side Case Spice

The agency rigs up a 20 m zipline escape route from the HQ roof. The line meets the flat ground at an angle of 32°. How far from the base of the building does it anchor? Round to two decimal places.

Before you solve: sketch it first. The story is new, but the three side names are old suspects.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Stand at the 32°. The zipline is the slanted side, so it is the hypotenuse (20 m). The anchor distance runs along the ground NEXT TO your angle. Adjacent and hypotenuse: which informant is that?

🔍 Show me a smaller case

Warm up: a 10 m zipline at 60°. How far along the ground? Adjacent and hypotenuse means cos: distance = 10 × cos 60° = 10 × 0.5 = 5 m.

Same structure. Now do your real case with 20 and 32°.

✅ Show the full solution

Decode. Angle = 32°. Hypotenuse = 20 m (the zipline). Want = adjacent (the anchor distance), call it a. Adjacent + hypotenuse = cos (CAH).

cos 32° = a ÷ 20

The unknown is on top of the fraction, so multiply both sides by 20: a = 20 × cos 32°.
Calculator (degrees mode!): cos 32° ≈ 0.8480.
a = 20 × 0.8480481 = 16.9609...
Round: a ≈ 16.96 m.

🌶️ The trap The last two wall problems used sin, so the hand reaches for sin out of habit: 20 × sin 32° = 10.60 m. That number is real, but it is the HEIGHT of the roof, not the anchor distance. The informant follows the sides in the question, never the story.

👃 Sniff test The ground run must be shorter than the zipline itself, and 16.96 is less than 20. At a shallow 32° the line should cover most of its length along the ground, and 16.96 out of 20 is most of it. It makes sense.

Case closed ✓

📓 Case Notes

Which two sides told you it was cos?
What answer would sin have given here, and what is that number actually measuring?
Say the two decisions (informant, then move) in order.

CASE T6 🗼 Upside-down Case Spice

The agency lookout tower is 18 m tall. From where Detective Kelcy stands on flat ground, the sight line up to the top makes an angle of 34° with the ground. How far is she from the base of the tower? Round to two decimal places.

The unknown is on the bottom of the fraction again, like the ramp case, but a different informant this time.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

The tower (18 m) is across from your angle, and the distance you want runs along the ground beside it. No hypotenuse anywhere in this question. Opposite and adjacent: who is that, and where does the unknown sit in the fraction?

🔍 Show me a smaller case

Warm up: a 5 m post, sight line at 45°. How far away are you? tan 45° = 5 ÷ d, so d = 5 ÷ tan 45° = 5 ÷ 1 = 5 m.

Equal angle, equal sides, feels right. Now do your real case with 18 and 34°.

✅ Show the full solution

Decode. Angle = 34°. Opposite = 18 m (the tower). Want = adjacent (the distance along the ground), call it d. Opposite + adjacent = tan (TOA).

tan 34° = 18 ÷ d

The unknown is on the bottom, so swap it with the trig value: d = 18 ÷ tan 34°.
Calculator: tan 34° ≈ 0.6745.
d = 18 ÷ 0.6745085 = 26.6860...
Round: d ≈ 26.69 m.

🌶️ The trap 18 × tan 34° = 12.14 m out of habit. But 34° is shallower than 45°, so she must be standing FURTHER from the tower than the tower is tall, and 12.14 fails that test before the calculator cools down. Multiply on top, divide on the bottom.

👃 Sniff test An angle under 45° means the adjacent beats the opposite, and 26.69 is more than 18. Check it forwards: 26.686 × tan 34° lands right back on 18.00. It makes sense.

Case closed ✓

📓 Case Notes

What is the tell that this is a divide case?
Why did the 45° line matter for sanity-checking?
This case had no hypotenuse at all: what does that instantly tell you?

CASE T7 🛝 Mystery-Angle Case Spice

The agency escape slide is 5.5 m long and its base runs 4.8 m along the floor. The safety file says a slide steeper than 30° needs a permit. What angle does the slide make with the floor, to the nearest degree, and does it pass?

Two sides known, no angle given anywhere. What kind of case walks in like that?

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

The 4.8 m base touches the mystery angle, and the 5.5 m slide is the slanted side. Adjacent and hypotenuse is cos, but cos θ = 0.87 does not finish the job: you need the un-cos button, cos⁻¹ (SHIFT then cos).

🔍 Show me a smaller case

Warm up: a 2 m plank leans with its top 1 m up the wall. What is its angle? sin θ = 1 ÷ 2 = 0.5, so θ = sin⁻¹(0.5) = 30°.

The unlock move is identical for all three informants. Now do your real case with 4.8 and 5.5.

✅ Show the full solution

Decode. Adjacent = 4.8 m (the base). Hypotenuse = 5.5 m (the slide). Want = the angle θ. Adjacent + hypotenuse = cos (CAH).

cos θ = 4.8 ÷ 5.5 = 0.8727...

The angle is locked inside the cos. Unlock it with the inverse: θ = cos⁻¹(0.8727273).
Calculator: SHIFT, cos, then the fraction. Result: 29.2228...°
Round: θ ≈ 29°. Under 30°, so it passes the safety file, by a whisker.

🌶️ The trap Divide the wrong way (5.5 ÷ 4.8 = 1.1458) and cos⁻¹ throws a Math ERROR. That error is a clue, not a disaster: sin and cos can never beat 1, because the hypotenuse is always the longest side and always on the bottom. An error here means your fraction is upside down.

👃 Sniff test The base is nearly as long as the slide, so the angle should be smallish, well under 45°, and 29° is. Check it forwards: cos 29° ≈ 0.8746, close to 0.8727. It makes sense.

Case closed ✓

📓 Case Notes

How do you spot a Mystery-Angle Case before touching the calculator?
What is your calculator's Math ERROR actually telling you?
Which of the three inverse buttons have you now used, and what does each undo?

Goblin translation: looking up, looking down

You already interrogated an angle of elevation in CASE T6, it just was not wearing its badge. When a question talks about looking up or looking down, here is the translation.

The words	What they mean	Where the angle lives
Angle of elevation	Looking UP at something from the flat	Between the HORIZONTAL and your sight line
Angle of depression	Looking DOWN at something from up high	Between the HORIZONTAL and your sight line
"From the horizontal"	Both angles start at the flat line	Never measured from a wall or cliff face

The two 15° angles are twins. Parallel flat lines plus one sight line make a Z, and the Z's corners match.

🧌

Goblin says: the depression angle is measured OUTSIDE the triangle, up at the horizontal. You never use it where it is: you copy it to the far corner (its alternate-angle twin) and work there.

🌶️ The trap The classic crime is measuring depression from the wall of the cliff instead of the horizontal. If your working uses 75° where the question said 15°, this is why. Draw the horizontal FIRST, every time.

CASE T8 🚣 Look-down Case Spice

From the top of the 24 m harbour watchtower, a goblin sentry spots a suspect rowing away. The angle of depression to the rowboat is 15°. How far is the boat from the base of the tower? Round to two decimal places.

Draw the horizontal line at the TOP first. The 15° lives between that line and the sight line, not inside the tower.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

Copy the 15° down to its alternate-angle twin at the boat. Now it is an ordinary triangle: the tower (24 m) is opposite the boat's angle, and the distance along the water is adjacent to it. Opposite and adjacent, unknown on the bottom.

🔍 Show me a smaller case

Warm up: a 10 m cliff, angle of depression 45° to a dinghy. How far out is it? Twin the angle down: tan 45° = 10 ÷ d, so d = 10 ÷ 1 = 10 m.

Steep look-down, boat close in. Now do your real case with 24 and 15°.

✅ Show the full solution

Decode. Depression of 15° at the top = elevation of 15° at the boat (alternate angles). Opposite = 24 m (the tower). Want = adjacent (distance along the water), call it d. Opposite + adjacent = tan (TOA).

tan 15° = 24 ÷ d

The unknown is on the bottom, so swap it with the trig value: d = 24 ÷ tan 15°.
Calculator: tan 15° ≈ 0.2679.
d = 24 ÷ 0.2679492 = 89.5692...
Round: d ≈ 89.57 m.

🌶️ The trap Two ways to sink this case. Measuring from the tower wall means using 75°: 24 ÷ tan 75° = 6.43 m, a boat nearly under the tower, absurd for a shallow look-down. Or leaving the 15° at the top and pairing it with the wrong sides. Twin the angle down FIRST, then it is a case you have already cracked.

👃 Sniff test A shallow 15° look-down means the boat is far out compared with the tower height, and 89.57 m is nearly four towers away. Cross-check by the other route: the angle inside the top corner is 90° − 15° = 75°, and 24 × tan 75° = 89.5692, the same answer from the other end. It holds.

Case closed ✓

📓 Case Notes

Where does the depression angle actually live on your diagram?
What is the twin move called, and why is it legal?
Which wrong answer would the sniff test have caught, and how fast?

CASE T9 🚩 Double-Triangle Case Spice

The goblin flag flies on a pole on top of HQ. From a manhole 40 m from the base of the building, the angle of elevation to the BOTTOM of the flagpole is 35°, and to the TOP of the flagpole it is 42°. How tall is the flagpole itself? Round to two decimal places.

One diagram, two triangles, one shared bottom side. Draw before you press anything.

👀 Stuck? Knock three times

Knock, knock, knock: What am I doing? Why? Is it helping?

You cannot find the pole in one move, but you CAN find two full heights: ground to pole-top, and ground to pole-bottom. Both are tan cases from the same 40 m. What do you do with the two answers?

🔍 Show me a smaller case

Warm up: from 10 m away, the elevation to the bottom of an aerial is 30° and to its top is 45°. How tall is the aerial? Top height = 10 × tan 45° = 10 m. Bottom height = 10 × tan 30° = 5.7735 m. Aerial = 10 − 5.7735 = 4.2265 ≈ 4.23 m.

Two triangles, one subtraction. Now do your real case with 40, 35° and 42°.

✅ Show the full solution

Decode. Adjacent = 40 m, shared by both triangles. Height to the pole top: H = 40 × tan 42°. Height to the pole bottom: h = 40 × tan 35°. Flagpole = H − h.

Calculator: tan 42° ≈ 0.9004, so H = 40 × 0.9004040 = 36.0162.
Calculator: tan 35° ≈ 0.7002, so h = 40 × 0.7002075 = 28.0083.
Subtract the heights: 36.0162 − 28.0083 = 8.0079.
Round: flagpole ≈ 8.01 m. (Keep the full calculator values until the final subtraction.)

🌶️ The trap The tempting shortcut is subtracting the angles first: 42° − 35° = 7°, then 40 × tan 7° = 4.91 m. Wrong, and not by a little. tan does not share out over subtraction, and the angles are not heights. Subtract the HEIGHTS, never the angles.

👃 Sniff test Ground to the pole base is 28 m, so an 8 m pole on top is believable for a flag you can see from the street. Add it back: 28.0083 + 8.0079 = 36.0162, exactly the pole-top height. And the wrong-path 4.91 would mean the tan-of-difference trick worked, which the check just disproved. It holds.

Case closed ✓

📓 Case Notes

Why could this not be done in one move?
What exactly is wrong with subtracting the angles?
What is your personal rule now for when rounding is allowed to happen?

Rapid response drill: which informant would you call?

🧌

Goblin says: naming the right informant FAST is the real exam skill. The sums are the easy part; the call is the case.

No solving allowed. Fifteen seconds a case: name the two sides in play, call the informant, name the move (multiply, divide or inverse). The deciding IS the skill. This drill is also your warm-up for every trig session from now on, and on a tired day you can run this and nothing else.

A 12 m rope slide meets the ground at 50°. Want: the height of its top.
A ramp must rise at 20° and its base can only run 7 m along the floor. Want: the length of the ramp board.
A goblin stands 15 m from a tree, sight line to the top at 38°. Want: the tree's height.
A 9 m guy wire runs from the ground to the top of a 6 m pole. Want: the wire's angle with the ground.
A drone hovers 30 m up, and a spotlight beam must hit it at 55° from the ground. Want: how far from the point below the drone to stand.
A 5 m ladder has its foot 1.4 m out from the wall. Want: the ladder's angle with the ground.

✅ Check your calls

sin, multiply. Opposite (the height) from the hypotenuse (the 12 m slide), unknown on top.
cos, divide. Adjacent (7 m) and hypotenuse (the board), and the unknown is on the bottom of the fraction.
tan, multiply. Opposite (the height) from the adjacent (15 m), unknown on top.
sin⁻¹. Opposite (6) and hypotenuse (9) both known, and the angle is locked inside.
tan, divide. Opposite (30 m) known, adjacent wanted, unknown on the bottom.
cos⁻¹. Adjacent (1.4) and hypotenuse (5) both known, and the angle is locked inside.

👃 Sniff test Six calls, all three informants used, all three moves used. If any call surprised you, it is telling you which case TYPE to reopen: a surprise multiply or divide sends you back to T5 or T6, a surprise inverse to T7.

Drill complete ✓

Cracked all nine? The triangle crimes unit salutes you.

Archive them in your case journal, and tell the goblin which triangle crime you want next. Rumour says the next unsolved file involves a compass and a very lost goblin.