Quarry School

The special angles: 30°, 45° and 60°

Explain it like I am five

Picture a yield sign, a triangle with three equal sides, each 2 feet. Its equal angles share 180°, so each is 60°. Fold it in half through one corner: the far side splits into 1 and 1, that corner's 60° into 30° and 30°. Each half is a right triangle with hypotenuse 2, short leg 1 and long leg 22−12 = 3 ≈ 1.73 (3 is the number that times itself makes 3).

Stand at the 30° corner. The short leg, 1, is across from you, so sin 30° = 12. The long leg, 3, touches you, so cos 30° = 32. At the 60° corner the legs swap: sin 60° = 32 and cos 60° = 12.

For 45°, fold a square napkin with sides 1 corner to corner. Each half has legs 1 and 1 and a diagonal of 12+12 = 2, so sin 45° = cos 45° = 12. Multiplying top and bottom by 2 gives 22 ≈ 0.707. These are the special angles: their values come out exact.

In plain words

Picture cutting a square sandwich along its diagonal. Each half is a right triangle with two equal legs and two 45° corners. A second useful shape comes from cutting an equilateral triangle, a triangle with three equal sides, down the middle. Each half has a 30° corner and a 60° corner. These are the two special triangles. They give exact values, answers with fractions and square roots, for the special angles 30°, 45° and 60°. In radians those angles are π6, π4 and π3. You can rebuild their values by drawing the triangles and reading the side ratios. You do not need six unrelated lists.

30°60°√31245°45°11√2
Half a square gives sides 1, 1, 2; half an equilateral triangle gives 1, 3, 2, with the short side facing 30°.
Reminder
  • SOH-CAH-TOA. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent. With opposite 4 and hypotenuse 8, sine is 48 = 12.
  • Rationalizing. 13 × 33 = 33; multiplying by a fraction equal to 1 preserves the value.
  • Coterminal angles. A full turn ends on the same ray: 405° − 360° = 45°. The same ray gives the same triangle ratios.
Why it works. A square with side 1 has a diagonal of 12+12 = 2. That diagonal halves two square corners, making 45° angles. An equilateral triangle with sides 2 has three equal angles: 180° ÷ 3 = 60° each. Its middle cut halves the base into lengths 1 and 1, and halves the top corner into two 30° angles. Each half therefore has hypotenuse 2 and long leg 22−12 = 3. The short leg faces 30°. Enlarging either triangle multiplies every length by the same number, which cancels from every side ratio.
Rule45°, 45°, 90°: sides 1, 1, 2. 30°, 60°, 90°: the side across from 30° is 1, across from 60° is 3, across from 90° is 2. sin 30° = 12, sin 45° = 22, sin 60° = 32; the cosines are the same list in reverse order. For an angle that makes full turns and then ends on an acute ray, we define its trig values to be those of that acute direction; this extends the triangle definition using the terminal side.
The same idea, five ways
Say it

Say thirty degrees, forty-five degrees and sixty degrees. Say root two over two for 22.

Write it

The side ratios of the two special triangles give exact values for three special angles.

In math
  • 30° = π6, 45° = π4, 60° = π3
  • 45° triangle: 1, 1, 2
  • 30° triangle: opposite 1, adjacent 3, hypotenuse 2
Like

Two sandwich cuts give reusable triangle patterns.

See it
30°60°√31245°45°11√2
Match the named angle with the side across from it before writing a ratio.
The same idea, other ways
As two cut shapes

A diagonal cuts a square into the equal-leg triangle. A middle cut splits an equilateral triangle into the short-leg and long-leg triangle. Keep those two pictures; their side lengths rebuild the reference table.

30°60°√31245°45°11√2
The short leg faces 30°; equal legs face the two 45° angles.
As a sine ladder

For the three acute special angles, place 1, 2 and 3 over 2 in increasing angle order. Read the three results in the table. This memory device is for these named angles; it is not a formula for other angles.

input θoutput sin θ30°[[1|2]]45°[[√{2}|2]]60°[[√{3}|2]]
Read the sine under each angle; the square-root number increases by one.
As exchanging your viewpoint

The 30° and 60° corners share one triangle. When you walk to the other corner, opposite and adjacent exchange places. Therefore sin 30° = cos 60° and sin 60° = cos 30°. In the equal-leg triangle sine and cosine agree.

θ√{3}12
Stand at the 30° corner, then exchange the two legs to stand at 60°.
θradianssin θcos θtan θcot θsec θcsc θ
30°π612323332332
45°π422221122
60°π332123332233
.1The equal-leg 45°, 45°, 90° triangle

Cut a square diagonally, from one corner to the opposite corner. The two perpendicular sides become equal legs. From either acute corner, opposite and adjacent have the same length, so sine and cosine match.

  • Rule: the side proportions are 1, 1 and 2.
  • Rule: equal opposite and adjacent legs give tan = cot = 1.
  • Rule: sin = cos = 22 and sec = csc = 2.
θ11√{2}
The equal legs make equal sine and cosine ratios.
Worked exampleRead cosine from an enlarged equal-leg triangle

An acute angle γ has adjacent leg 6, opposite leg 6 and hypotenuse 62. Find cos γ. In words, compare the touching leg with the longest side.

θ666√{2}
The cosine compares the leg 6 with the hypotenuse 62.
  1. cos γ = 662 = 12.Cosine is adjacent over hypotenuse; divide top and bottom by 6.
  2. 12 = 22.Multiply top and bottom by 2, which squares to 2.
Answer
cos γ = 22
Check Both legs are equal, so sin γ has the same value; their squared values add to 12 + 12 = 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The hypotenuse is 2 because the legs are each 1.
Lengths do not add to form the hypotenuse.
✓ Instead: The squares add: 12 + 12 = 2, so hypotenuse = 2.
Tips and tricks
  • Equal legs mean equal sine and cosine.
.2The short-leg 30°, 60°, 90° triangle

Split an equilateral triangle down the middle. The 30° corner faces the short leg. The 60° corner faces the long leg. Stand at the requested corner before deciding which length goes on top.

  • Rule: short leg, long leg and hypotenuse have proportions 1, 3 and 2.
  • Rule: the short leg faces 30°; the long leg faces 60°.
  • Rule: exchanging these two corners exchanges sine and cosine.
θ√{3}12
From 30°, the short leg is opposite and the long leg is adjacent.
Worked exampleName the corner before using tangent

An acute angle β has opposite leg 3, adjacent leg 33 and hypotenuse 6. Find tan β. In words, compare the facing leg with the touching leg.

θ3√{3}36
The short leg faces β, identifying the 30° corner.
  1. Divide all lengths by 3: opp = 1, adj = 3, hyp = 2.The common scale factor preserves every ratio.
  2. tan β = 333 = 13 = 33.Tangent uses opposite over adjacent; cancel the 3 and rationalize.
Answer
tan β = 33
Check cot β = 3, so tan β × cot β = 33 × 3 = 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: tan β = 3 for this short-leg angle.
That puts adjacent over opposite and gives cotangent.
✓ Instead: tan β = 33, while cot β = 3.
Tips and tricks
  • Write opp and adj beside the two legs before forming tangent.
.3Full turns followed by an acute direction

Think of a compass pointer spinning twice before it points northeast. Its final direction is unchanged. We extend the acute triangle definition by choosing trig values from the final direction, rather than from the number of full turns. When that direction is acute, use the right triangle at that angle. The point lesson will extend this choice to every direction.

  • For an acute final direction, removing full turns keeps all six trig values.
  • 765° − 360° − 360° = 45°, so 765° uses the 45° triangle.
45°765°same terminal side
Two full turns leave the same 45° terminal ray.
Reminder
  • Coterminal. Coterminal angles have the same terminal side because they differ by full turns.
The same idea, five ways
Say it

Full turns do not change a trig value when the final direction is the same acute ray.

Write it

We extend the acute-angle functions by assigning the same outputs to angles with that terminal side.

In math
  • sec 765° = sec 45° = 2
Like

A compass pointer gives the same direction after two complete spins.

See it
45°765°same terminal side
Two full turns leave the same 45° terminal ray.
Worked exampleTwo full turns before reading a ratio

Find sec 765° exactly. In words, remove two complete turns and compare the hypotenuse with the adjacent leg.

45°765°same terminal side
Two full turns leave the same 45° terminal ray.
  1. 765° − 360° = 405°; 405° − 360° = 45°.Each full turn keeps the terminal ray, and our extension assigns its acute-direction values.
  2. At 45°, adj = 1 and hyp = 2, so sec 765° = 21 = 2.Secant is hypotenuse over adjacent in the shared equal-leg triangle.
Answer
sec 765° = 2
Check Its cosine is 22, and 2 × 22 = 22 = 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Two extra full turns multiply the secant by 2.
Our extension reads the final ray, which the extra turns leave fixed; it does not count the turns.
✓ Instead: sec 765° = sec 45° = 2.
Tips and tricks
  • Reduce the angle first, then check whether the final ray is acute.
Strategy: step by step
  1. Sketch the triangle that contains your angle and write its three sides.
  2. Stand at your angle and name opp, adj and hyp.
  3. Read sin, cos and tan with SOH-CAH-TOA.
  4. Flip for csc, sec and cot, and rationalize.
  5. Use the sine reference diagram for the three acute special angles: each input and its exact output share one column. The axis lesson will add the two endpoint columns at 0° and 90°.
Strategy
Rebuild an exact special-angle value
1
Does the angle include extra full turns?
YesSubtract full turns until the same terminal side appears within one turn.
NoKeep the angle.
↓
2
Is the resulting acute angle 45°?
YesUse sides 1, 1 and 2.
NoFor 30° or 60°, use sides 1, 3 and 2 and name the legs from that corner.
↓
3
Does the answer have a square root on the bottom?
YesMultiply top and bottom by that root and reduce.
NoKeep the reduced exact value.
  1. Identify the special angle and choose its special triangle.
  2. For an extra full turn, remove that turn first: 390° − 360° = 30°. The same ray gives the same acute triangle.
  3. Label opposite, adjacent and hypotenuse from the selected corner.
  4. Use the function's side ratio, reduce, then rationalize its denominator.
  5. Check the result with a reciprocal product or the Pythagorean identity.
Worked exampleRead six ratios from a scaled special triangle

A right triangle has opposite leg 43, adjacent leg 4 and hypotenuse 8, with respect to β. Find all six functions. In words, turn these three lengths into six ratios.

θ44√{3}8
The scale factor cancels from each ratio.
θ44√{3}8
The scale factor cancels from each ratio.
input θoutput radians30°[[π|6]]45°[[π|4]]60°[[π|3]]
Reference table: read the radians entry directly under its θ label.
input θoutput sin θ30°[[1|2]]45°[[√{2}|2]]60°[[√{3}|2]]
Reference table: read the sin θ entry directly under its θ label.
input θoutput cos θ30°[[√{3}|2]]45°[[√{2}|2]]60°[[1|2]]
Reference table: read the cos θ entry directly under its θ label.
input θoutput tan θ30°[[√{3}|3]]45°160°√{3}
Reference table: read the tan θ entry directly under its θ label.
input θoutput cot θ30°√{3}45°160°[[√{3}|3]]
Reference table: read the cot θ entry directly under its θ label.
input θoutput sec θ30°[[2√{3}|3]]45°√{2}60°2
Reference table: read the sec θ entry directly under its θ label.
input θoutput csc θ30°245°√{2}60°[[2√{3}|3]]
Reference table: read the csc θ entry directly under its θ label.
  1. opp = 43, adj = 4, hyp = 8.The question identifies the legs relative to β.
  2. sin β = 438 = 32; cos β = 48 = 12.Sine uses opposite over hypotenuse; cosine uses adjacent over hypotenuse. Divide top and bottom by 4.
  3. tan β = 434 = 3; cot β = 443 = 13 = 33.Tangent and cotangent compare the legs in opposite orders.
  4. sec β = 84 = 2; csc β = 843 = 23 = 233.Use hypotenuse over adjacent and hypotenuse over opposite; rationalize the last denominator.
Answer
  • sin β = 32
  • cos β = 12
  • tan β = 3
  • cot β = 33
  • sec β = 2
  • csc β = 233
Check (43)2 + 42 = 48 + 16 = 64 = 82. Dividing every length by 4 gives the special triangle with lengths 3, 1 and 2.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: one ratio from equal legs

An acute angle has opposite leg 3, adjacent leg 3 and hypotenuse 32. Find its sine. In words, divide the facing length by the longest length.

θ333√{2}
The common factor 3 does not change the ratio.
  1. sin = 332 = 12.Sine is opposite over hypotenuse, and the 3 cancels.
  2. 12 = 22.Multiply top and bottom by 2.
Answer
sin = 22
Check Cosine is identical because the legs agree; their squares total 1.
Rung 2Rung 2: a full turn before a reciprocal

Find cot 390° exactly. In words, remove a full turn and compare adjacent with opposite.

30°390°same terminal side
One extra lap keeps the 30° terminal ray.
  1. 390° − 360° = 30°.A full turn returns to the same ray, so the acute triangle ratios agree.
  2. For that corner, opp = 1, adj = 3, hyp = 2.The shortest leg faces the 30° corner.
  3. cot 390° = 31 = 3.Cotangent is adjacent over opposite.
Answer
cot 390° = 3
Check tan 390° = 33, and their product is 1.
Rung 3Rung 3: rationalize an equal-leg reciprocal

Find sec 405° exactly. In words, find hypotenuse over adjacent after removing a full turn.

45°405°same terminal side
The final ray is the equal-leg triangle's ray.
  1. 405° − 360° = 45°.A full turn ends on the same ray, so use the equal-leg triangle.
  2. cos 405° = 12 = 22.Adjacent over hypotenuse is 1 over 2; multiply top and bottom by 2.
  3. sec 405° = 1 ÷ 22 = 22 = 222 = 2.Secant flips the nonzero cosine; rationalize with 2 and reduce.
Answer
sec 405° = 2
Check cos 405° = 22, so their product is 22 = 1.
Rung 4Rung 4: all six after a full turn

Find all six functions of 420° exactly. In words, return to the same acute ray and write the six side ratios.

60°420°same terminal side
The long leg faces the final 60° ray.
  1. 420° − 360° = 60°.One full turn changes neither the final ray nor its side ratios.
  2. opp = 3, adj = 1 and hyp = 2.At 60° the long leg is across from the corner.
  3. sin = 32, cos = 12, tan = 3.Apply SOH-CAH-TOA.
  4. cot = 13 = 33, sec = 2, csc = 23 = 233.Flip the first three ratios and rationalize root denominators.
Answer
  • sin 420° = 32
  • cos 420° = 12
  • tan 420° = 3
  • cot 420° = 33
  • sec 420° = 2
  • csc 420° = 233
Check sin2420° + cos2420° = 34 + 14 = 1.
Rung 5Rung 5: combine two exact products

Find cos 390° sin 420° + sin 390° cos 420° exactly. In words, evaluate the four functions, multiply each pair and add.

390° shares the 30° ray
420° shares the 60° ray
34 + 14 = 1
Evaluate the pairs before adding them.
  1. 390° shares the 30° ray; 420° shares the 60° ray.Subtract 360° from each angle before reading its triangle.
  2. cos 390° sin 420° = 32 × 32 = 34.The root factors multiply to 3.
  3. sin 390° cos 420° = 12 × 12 = 14.Multiply fraction tops and bottoms.
  4. 34 + 14 = 44 = 1.The bottoms match, so add the tops.
Answer
1
Check Exchanging the 30° and 60° corners, which add to 90°, makes the expression sin2420° + cos2420°, which equals 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: sin 30° = 32
The long leg faces 60°, not 30°. The special angle decides which leg is opposite.
✓ Instead: sin 30° = 12; sin 60° = 32.
✗ Not this: The sine ladder says sin 40° = 22.
The three entries correspond to three named angles; 40° is absent. The pattern is not a formula for every angle.
✓ Instead: Use the exact table for its special angles; use a suitable other method or a calculator for 40°.
✗ Not this: Multiplying every triangle side by 4 multiplies its sine by 4.
Both the top and bottom of the ratio multiply by 4, so the scale factor cancels.
✓ Instead: 438 = 32.
Tips and tricks
  • Sketch the two special triangles first. Rebuild cot, sec and csc by flipping their partners.
  • The shortest leg faces the smallest angle. This keeps the 30° and 60° values in order.
  • The sine ladder's cosines run backward; use that as a check after identifying the angle.
  • Read reference tables by columns: keep the input label and its output in the same vertical column.
  • The special values check the earlier warning: sin 45° + cos 45° = 2 ≈ 1.414, not 1.
Trap. Swapping the 30° and 60° values. The shortest side, 1, faces the smallest angle, 30°, so sin 30° = 12 is the small sine and sin 60° = 32 (about 0.866) is the large one. From 0° to 90°, sine grows as the angle grows.
Keep in mind
  • The shortest side faces the smallest angle: 1 faces 30°, 3 ≈ 1.73 faces 60°, and 2 faces the 90° corner.
  • The cosines are the sines read backward: cos 30° = sin 60° = 32 and cos 60° = sin 30° = 12.
  • 13 and 33 are the same number, about 0.577, and multiplying top and bottom by 3 turns the first into the rationalized second.
  • On your cheat sheet, draw the two triangles with sides 1, 1, 2 and 1, 3, 2, and label the angles π6, π4, π3 in radians too: every value rebuilds from them.
Memory hookCount 0 to 4 under the root: the sines of 0°, 30°, 45°, 60°, 90° are 02, 12, 22, 32, 42. Cosine reads the list backward.
Flash cards: say the answer out loud, then flip
What are the sides of the 45°, 45°, 90° triangle?
1, 1 and 2
What are the sides of the 30°, 60°, 90° triangle, and which side faces 30°?
1, 3 and 2. The side 1 faces 30°.
A 12-foot ladder leans at 30° to the floor. How high up the wall does it reach?
12 × sin 30° = 12 × 12 = 6 feet
A square has sides 5. How long is its diagonal?
52 ≈ 7.07, the 45° triangle scaled by 5
What are 30°, 45° and 60° in radians?
π6, π4 and π3
Is sin 60° = 12?
No. 12 is sin 30°. The short side 1 faces the small angle, so sin 60° = 32 ≈ 0.866.