A two-turn compass reveals sine, cosine and tangent angle-sum and angle-difference formulae, exact 75° and 15° values and valid interval solutions.
Explore the compound-angle guide
- The quick answer: one 75° direction, three exact values
- Begin with a unit-length compass pointer
- First turn: rotate to 45°
- What a second rotation does to coordinates
- The cosine addition formula
- Derive sin 75° exactly
- Derive cos 75° exactly
- Derive tan 75° from the exact sine and cosine
- The tangent addition formula gives the same answer
- Verify the unit pointer still has length one
- Make the second turn clockwise instead
- Find sin 15° exactly
- Find cos 15° exactly
- Find tan 15° and compare
- The tangent values are reciprocal in this case
- Compare 15° and 75° in one table
- Why sin(A+B) is not sin A plus sin B
- Why cos(A+B) is not cos A plus cos B
- Why an angle sum does not simply double a coordinate
- A pointer of length two provides an applied calculation
- A compound-angle expression can be simplified symbolically
- Rewrite the corresponding cosine expression
- Check the symbolic expansion at x = 0°
- Solve a shifted-angle equation in an interval
- Verify both shifted-angle solutions
- A second interval changes the answer set
- Use radians without changing the identities
- A subtraction formula for tangent must retain its denominator
- A quick domain example for tangent addition
- Choose the most useful representation
- Common mistake: using the wrong cosine sign
- Common mistake: reversing sine and cosine terms
- Common mistake: treating tangent as a simple sum
- Common mistake: rounding surds too early
- Common mistake: forgetting the transformed interval
- A six-minute independent check
- What parents can ask without learning compound-angle notation
- How small-group teaching can repair this concept
- Where this sits in Singapore G3 Additional Mathematics
- Frequently asked questions
- Continue through the Additional Mathematics series
- Official curriculum and model boundaries
What is Additional Mathematics when one direction is reached by adding two familiar angles? Imagine a compass pointer first turning 45° anticlockwise, then turning another 30° in the same direction. Its final direction is 75° from the starting axis, but the sine, cosine and tangent of 75° are not obtained by simply adding the corresponding trigonometric values.
This is the purpose of compound-angle formulae. They describe exactly how the trigonometric components change when angles are added or subtracted. These identities appear under Geometry and Trigonometry G1 in Singapore’s 2027 G3 Additional Mathematics syllabus K341.
The two-turn compass is a mathematical illustration, not a real navigational instrument. Its pointer and coordinates are idealised, angles are measured from a chosen horizontal axis, and every numerical result can be checked using exact special-angle values.
The quick answer: one 75° direction, three exact values
Start with 45°, add 30° and obtain a total direction of 75°.
The compound-angle identities give:
sin 75° = (√6 + √2)/4.
cos 75° = (√6 − √2)/4.
tan 75° = 2 + √3.
Their approximate values are:
sin 75° ≈ 0.9659.
cos 75° ≈ 0.2588.
tan 75° ≈ 3.7321.
None requires a calculator to establish the exact expression. We derive each using the known trigonometric values of 30° and 45°.
The exact forms are useful because they preserve algebraic relationships that early rounding can obscure.
Begin with a unit-length compass pointer
Imagine the centre of the compass is at the origin O = (0, 0).
Let the pointer have length one.
A direction angle θ measured anticlockwise from the positive x-axis places its tip at:
P(θ) = (cos θ, sin θ).
The first coordinate is the horizontal component of the unit pointer.
The second is its vertical component.
If θ = 0°, the pointer points to (1, 0).
If θ = 90°, it points to (0, 1).
A compound angle describes what happens when we rotate a direction by a further angle, not when we make the pointer longer.
First turn: rotate to 45°
After an anticlockwise turn of 45°, the tip lies at:
P(45°) = (cos 45°, sin 45°).
The special-angle values are:
cos 45° = √2/2.
sin 45° = √2/2.
Therefore:
P(45°) = (√2/2, √2/2).
Both components are positive and equal, consistent with a direction halfway between the positive x-axis and the positive y-axis.
The compass now makes another 30° turn.
The final angle will be 75°.
The question is how the existing horizontal and vertical components change under that second rotation.
What a second rotation does to coordinates
A plane rotation through angle B transforms coordinates (x, y) into:
x_new = x cos B − y sin B.
y_new = x sin B + y cos B.
These expressions are the algebraic effects of rotating a vector without changing its length.
If the starting vector is the unit direction:
(x, y) = (cos A, sin A),
then the new vertical coordinate becomes:
y_new = cos A sin B + sin A cos B.
But the final direction is A + B, whose vertical component is:
sin(A + B).
Therefore:
sin(A + B) = sin A cos B + cos A sin B.
This is the sine addition formula.
The identity follows from a coordinate rotation, rather than an arbitrary sign pattern.
The cosine addition formula
Use the rotated horizontal coordinate:
x_new = x cos B − y sin B.
Substitute:
x = cos A.
y = sin A.
Then:
x_new = cos A cos B − sin A sin B.
The final direction A + B has horizontal coordinate cos(A + B).
Therefore:
cos(A + B) = cos A cos B − sin A sin B.
Notice the minus sign.
It reflects how the original vertical component contributes negatively to the new horizontal component during anticlockwise rotation.
The sine formula uses a plus between its two products.
The cosine formula uses a minus.
Understanding where those signs come from reduces memorisation errors.
Derive sin 75° exactly
Write:
75° = 45° + 30°.
Use:
sin(A + B) = sin A cos B + cos A sin B.
Substitute A = 45° and B = 30°:
sin 75° = sin 45° cos 30° + cos 45° sin 30°.
Use exact values:
sin 45° = √2/2.
cos 30° = √3/2.
cos 45° = √2/2.
sin 30° = 1/2.
Therefore:
sin 75° = (√2/2)(√3/2) + (√2/2)(1/2).
Multiply the fractions:
sin 75° = √6/4 + √2/4.
Combine:
sin 75° = (√6 + √2)/4.
The result is positive and less than one, as expected for a first-quadrant sine.
Derive cos 75° exactly
Use:
cos(A + B) = cos A cos B − sin A sin B.
Substitute:
cos 75° = cos 45° cos 30° − sin 45° sin 30°.
Use the exact special-angle values:
cos 75° = (√2/2)(√3/2) − (√2/2)(1/2).
Therefore:
cos 75° = √6/4 − √2/4.
So:
cos 75° = (√6 − √2)/4.
This value is positive but relatively small, which fits the compass pointing close to the vertical axis.
The vertical component sin 75° is large, while the horizontal component cos 75° is small.
Both describe a unit-length pointer at the same angle.
Derive tan 75° from the exact sine and cosine
By definition, where cosine is nonzero:
tan θ = sin θ/cos θ.
Therefore:
tan 75° = [(√6 + √2)/4]/[(√6 − √2)/4].
The factors of four cancel:
tan 75° = (√6 + √2)/(√6 − √2).
Multiply numerator and denominator by the conjugate √6 + √2:
tan 75° = (√6 + √2)²/(6 − 2).
Expand the numerator:
(√6 + √2)² = 6 + 2√12 + 2 = 8 + 4√3.
Thus:
tan 75° = (8 + 4√3)/4.
Therefore:
tan 75° = 2 + √3.
The tangent is greater than one because the pointer has a greater vertical component than horizontal component.
The tangent addition formula gives the same answer
Another prescribed compound-angle identity is:
tan(A + B) = [tan A + tan B]/[1 − tan A tan B],
where the relevant tangent values and denominator are defined.
For A = 45° and B = 30°:
tan 45° = 1.
tan 30° = 1/√3.
So:
tan 75° = [1 + 1/√3]/[1 − 1/√3].
Multiply numerator and denominator by √3:
tan 75° = (√3 + 1)/(√3 − 1).
Rationalise the denominator:
tan 75° = (√3 + 1)²/(3 − 1).
Expand:
(3 + 2√3 + 1)/2 = (4 + 2√3)/2.
This gives:
tan 75° = 2 + √3.
The result agrees with the sine-over-cosine calculation.
Verify the unit pointer still has length one
Rotating a unit vector does not change its length.
Therefore the exact coordinates must satisfy:
sin² 75° + cos² 75° = 1.
Substitute:
[(√6 + √2)/4]² + [(√6 − √2)/4]².
The first numerator square is:
8 + 4√3.
The second numerator square is:
8 − 4√3.
Add them and divide by sixteen:
[(8 + 4√3) + (8 − 4√3)]/16 = 16/16.
Therefore the sum is exactly one.
This independent identity confirms that the two compound-angle results are consistent with a unit-length direction.
Make the second turn clockwise instead
Now suppose the pointer starts at 45° but turns 30° clockwise.
Its final direction is:
45° − 30° = 15°.
Subtraction formulae can be derived by replacing B with −B in the addition formulae.
Using:
sin(−B) = −sin B
and:
cos(−B) = cos B,
we obtain:
sin(A − B) = sin A cos B − cos A sin B.
Likewise:
cos(A − B) = cos A cos B + sin A sin B.
The signs differ from the addition versions in a predictable way.
Find sin 15° exactly
Write:
15° = 45° − 30°.
Use the sine subtraction formula:
sin 15° = sin 45° cos 30° − cos 45° sin 30°.
Substitute:
sin 15° = (√2/2)(√3/2) − (√2/2)(1/2).
Therefore:
sin 15° = (√6 − √2)/4.
This is the same exact expression we obtained for cos 75°.
That is expected because 15° and 75° are complementary angles:
sin 15° = cos 75°.
The relationship provides a check on the subtraction formula.
Find cos 15° exactly
Use:
cos 15° = cos 45° cos 30° + sin 45° sin 30°.
Substitute the exact values:
cos 15° = √6/4 + √2/4.
Therefore:
cos 15° = (√6 + √2)/4.
This matches sin 75°.
The angle pair is complementary:
cos 15° = sin 75°.
These relationships are not coincidences. They follow from the geometry of a right angle and from the compound-angle formulae.
The exact expressions switch positions when the pointer moves from 75° to 15°.
Find tan 15° and compare
Use the tangent subtraction formula:
tan(A − B) = [tan A − tan B]/[1 + tan A tan B].
For A = 45°, B = 30°:
tan 15° = [1 − 1/√3]/[1 + 1/√3].
Simplify:
tan 15° = (√3 − 1)/(√3 + 1).
Rationalise:
tan 15° = (√3 − 1)²/(3 − 1).
The numerator expands to:
3 − 2√3 + 1 = 4 − 2√3.
Divide by two:
tan 15° = 2 − √3.
This is a positive small number, consistent with the pointer lying close to the horizontal axis.
The tangent values are reciprocal in this case
We found:
tan 75° = 2 + √3.
tan 15° = 2 − √3.
Multiply:
(2 + √3)(2 − √3) = 4 − 3 = 1.
Therefore the two tangent values are reciprocals.
This is consistent with the complementary-angle relationship:
tan(90° − θ) = 1/tan θ,
where both sides are defined.
The result provides another exact mathematical check.
It also demonstrates a useful pattern involving algebraic conjugates: expressions of the form a + b and a − b can multiply to remove a surd.
Compare 15° and 75° in one table
| Function | At 15° | At 75° |
|---|---|---|
| Sine | (√6 − √2)/4 | (√6 + √2)/4 |
| Cosine | (√6 + √2)/4 | (√6 − √2)/4 |
| Tangent | 2 − √3 | 2 + √3 |
The sine and cosine values exchange places because the angles add to 90°.
The tangent values are reciprocal.
A student can use the table for self-checking, but should know how to derive the entries from 45° and 30° rather than memorise six disconnected surd expressions.
The ability to reconstruct an exact result is more transferable than remembering one finished number.
Why sin(A+B) is not sin A plus sin B
One common false shortcut is:
sin(A + B) = sin A + sin B.
Test A = 45° and B = 30°.
The correct result is:
sin 75° ≈ 0.9659.
The false right-hand side is:
sin 45° + sin 30° ≈ 0.7071 + 0.5 = 1.2071.
A real sine output cannot exceed one.
So the proposed identity is false.
The proper addition formula contains cross-products:
sin A cos B + cos A sin B.
The cosine factors are essential.
Why cos(A+B) is not cos A plus cos B
A similar false shortcut is:
cos(A + B) = cos A + cos B.
Using the same angles:
cos 75° ≈ 0.2588.
But:
cos 45° + cos 30° ≈ 0.7071 + 0.8660 = 1.5731.
The values do not agree, and the proposed sum is greater than the allowed cosine range.
The correct cosine addition formula is:
cos A cos B − sin A sin B.
The product terms and the negative sign are essential.
Numerical counterexamples are useful, but the rotation derivation gives the general truth.
Why an angle sum does not simply double a coordinate
Adding 30° to 45° changes the direction from 45° to 75°.
The vertical component does not increase by sin 30° in a straight additive manner.
The original pointer has both horizontal and vertical components, and the second rotation mixes them.
That is why compound-angle formulae have two products.
Each previous component contributes to the new orientation.
This insight helps students understand the relationship between rotation, trigonometry and vector-like coordinates without needing to memorise a separate rule for every possible total angle.
A pointer of length two provides an applied calculation
Suppose the compass pointer has length two units instead of one.
At 75°, its horizontal projection is:
x = 2 cos 75°.
So:
x = (√6 − √2)/2,
approximately 0.518 units.
Its vertical projection is:
y = 2 sin 75°.
Therefore:
y = (√6 + √2)/2,
approximately 1.932 units.
The squared projections add to the squared pointer length:
x² + y² = 4.
This example demonstrates how a trigonometric identity can be used to calculate components of a direction with a non-unit length.
The model is geometric, not a claim about a real device’s performance.
A compound-angle expression can be simplified symbolically
Suppose a question gives:
sin(x + 30°).
Apply the addition formula:
sin(x + 30°) = sin x cos 30° + cos x sin 30°.
Use exact constants:
cos 30° = √3/2.
sin 30° = 1/2.
Therefore:
sin(x + 30°) = (√3/2)sin x + (1/2)cos x.
This works for every real angle x, not only x = 45°.
The result is an identity rather than a solution to an equation.
That is why it can be used to rewrite an expression containing a shifted angle.
Rewrite the corresponding cosine expression
Similarly:
cos(x + 30°) = cos x cos 30° − sin x sin 30°.
Substitute exact values:
cos(x + 30°) = (√3/2)cos x − (1/2)sin x.
The two formulae are related but not identical.
The sine expansion gives a positive cosine contribution.
The cosine expansion gives a negative sine contribution.
A student should check which function appears on the left and which angle operation appears inside the bracket.
A wrong sign may still yield a plausible-looking expression, so an exact test angle is valuable.
Check the symbolic expansion at x = 0°
For the sine expansion:
sin(0° + 30°) = sin 30° = 1/2.
The right side gives:
(√3/2)sin0° + (1/2)cos0° = 0 + 1/2.
They agree.
For the cosine expansion:
cos(0° + 30°) = cos30° = √3/2.
The right side gives:
(√3/2)cos0° − (1/2)sin0° = √3/2.
They also agree.
Testing x = 0° cannot by itself prove a general identity, but it is a quick way to detect obvious sign or coefficient errors.
The general proof comes from the compound-angle formulae.
Solve a shifted-angle equation in an interval
Consider:
sin(θ + 30°) = 1/2,
for:
0° ≤ θ ≤ 180°.
The unknown in the sine function is:
u = θ + 30°.
Translate the interval:
30° ≤ u ≤ 210°.
Within that interval, sine equals one-half at:
u = 30°
and:
u = 150°.
Now convert back to θ by subtracting 30°:
θ = 0°
or:
θ = 120°.
Both lie in the original interval.
The final answer is:
θ = 0° or 120°.
This illustrates why the transformed-angle interval must be considered before listing solutions.
Verify both shifted-angle solutions
For θ = 0°:
sin(θ + 30°) = sin 30° = 1/2.
For θ = 120°:
sin(θ + 30°) = sin 150° = 1/2.
Both satisfy the original equation.
The compound-angle identity could also be used to rewrite the equation as:
(√3/2)sin θ + (1/2)cos θ = 1/2.
But using the shifted angle directly may be simpler for this interval problem.
A-Math problem solving includes selecting the shortest correct method, not applying every formula available.
A second interval changes the answer set
Suppose the same equation is asked over:
0° ≤ θ < 90°.
Then:
30° ≤ θ + 30° < 120°.
Only u = 30° is in that transformed interval.
Therefore:
θ = 0°
is the only answer.
The equation has not changed.
The solution set has changed because the permitted interval has changed.
This is one reason the official syllabus specifies solving simple trigonometric equations in a given interval, rather than substituting an unrequested unrestricted solution formula.
Use radians without changing the identities
The same addition identities hold for angles in radians.
For example:
75° = 5π/12.
45° = π/4.
30° = π/6.
Therefore:
sin(5π/12) = sin(π/4 + π/6).
The exact result remains:
(√6 + √2)/4.
The angle notation changed, but the geometric direction did not.
However, a calculator in degree mode and one in radian mode interpret a bare numeric input differently.
Students should keep the input unit consistent throughout the working and final answer.
A subtraction formula for tangent must retain its denominator
The tangent subtraction identity is:
tan(A − B) = [tan A − tan B]/[1 + tan A tan B].
It is valid where the relevant tangent values are defined and the denominator is nonzero.
The addition identity uses:
1 − tan A tan B
in the denominator.
Students often remember the numerator but forget the denominator sign.
One way to reconstruct it is to write tangent as sine over cosine and substitute the compound-angle identities.
The fraction structure then follows from a general derivation, not an arbitrary memory trick.
A quick domain example for tangent addition
Take A = 45° and B = 45°.
The addition formula would give:
tan 90° = [1 + 1]/[1 − 1].
The denominator is zero.
That is correct: tan 90° is undefined.
The formula does not mean the tangent has an ordinary infinite real value.
It tells us the resulting direction is vertical, so the ratio of vertical to horizontal components requires division by zero.
A-Math students must not cancel or ignore such denominator restrictions.
Choose the most useful representation
For a numerical exact-value question such as sin 75°, the compound-angle expansion using 45° and 30° is direct.
For a simplification question such as sin(x + 30°), leaving the answer in terms of sin x and cos x may be the goal.
For an interval equation such as sin(θ + 30°) = 1/2, solving in the shifted angle may be shorter.
For a direction-projection problem, horizontal and vertical coordinates may be the final quantities.
The same identity supports all these tasks, but the requested output determines the most efficient final form.
A good learner knows why they are transforming an expression before beginning.
Common mistake: using the wrong cosine sign
The cosine addition formula has a minus between its two products:
cos(A + B) = cos A cos B − sin A sin B.
If the student uses plus, the 75° result becomes too large and fails the exact-value check.
Common mistake: reversing sine and cosine terms
The sine addition identity includes:
sin A cos B + cos A sin B.
It is not sin A sin B + cos A cos B.
The latter expression is cos(A − B), a different identity.
Recognising the geometry of rotation helps distinguish these forms.
Common mistake: treating tangent as a simple sum
In general:
tan(A + B) ≠ tan A + tan B.
The correct formula has a numerator and denominator.
For 45° plus 30°, the exact tangent is 2 + √3, not simply 1 + 1/√3.
Common mistake: rounding surds too early
Using decimals for √2 and √3 before completing the algebra can produce small rounding errors.
Exact surd forms show the identity clearly and preserve precision.
Approximate only when the question requests a decimal answer.
Common mistake: forgetting the transformed interval
For sin(θ + 30°), the interval for the sine argument is shifted by 30°.
Solve for all valid values of the argument before subtracting 30° to recover θ.
Otherwise valid solutions may be missed.
A six-minute independent check
1. State sin(A + B). 2. State cos(A + B). 3. Find sin 75° exactly using 45° + 30°. 4. Find cos 75° exactly. 5. Find tan 75° exactly. 6. Find sin 15° and cos 15° exactly. 7. Find tan 15° exactly. 8. Simplify sin(x + 30°). 9. Solve sin(θ + 30°) = 1/2 for 0° ≤ θ ≤ 180°. 10. Explain why tan(45° + 45°) is undefined.
Answers: sin A cos B + cos A sin B; cos A cos B − sin A sin B; (√6 + √2)/4; (√6 − √2)/4; 2 + √3; (√6 − √2)/4 and (√6 + √2)/4; 2 − √3; (√3/2)sin x + (1/2)cos x; θ = 0° or 120°; and the resulting angle is 90°, where cosine is zero.
The learner should derive at least one exact value rather than simply copy the finished list.
What parents can ask without learning compound-angle notation
Ask your child why a 45° turn followed by 30° gives a direction of 75°, but the two sine values cannot simply be added.
Ask why the pointer’s horizontal component gets smaller as it approaches vertical.
Ask what changes when the second turn is clockwise rather than anticlockwise.
Then ask why an interval question might have two different directions producing the same sine value.
These questions reveal whether the student understands the underlying geometry rather than memorising signs.
How small-group teaching can repair this concept
One student may know exact 30° and 45° values but reverse the signs in the cosine formula.
Another may derive sine and cosine correctly but struggle to rationalise the tangent quotient.
Another may calculate exact values but forget the angle interval while solving an equation.
A fourth may confuse degrees and radians in a calculator check.
The right lesson identifies the first incorrect reasoning step, repairs it and asks the learner to derive a different compound angle without copying the original answer.
The aim is accurate transfer from angle geometry to identities and equations.
Where this sits in Singapore G3 Additional Mathematics
The official 2027 Singapore–Cambridge Secondary Education Certificate G3 Additional Mathematics syllabus K341 explicitly includes expansions of sin(A ± B), cos(A ± B) and tan(A ± B), as well as exact special-angle values and solving simple trigonometric equations within a given interval.
These appear under Geometry and Trigonometry G1.
The two-turn compass is an original teaching metaphor connecting those prescribed mathematical relationships. It is not a real instrument or official examination question.
SEAB lists K341 as the 2027 G3 Additional Mathematics code, with 4049 as the earlier GCE O-Level reference for 2026 and before. G2 Additional Mathematics K232 has a separate syllabus.
Frequently asked questions
What is a compound-angle formula?
It is an identity that expresses the sine, cosine or tangent of an angle sum or difference in terms of the component angles.
Is sin(A + B) equal to sin A + sin B?
No, not generally. The correct formula is sin A cos B + cos A sin B.
Why does the cosine addition formula contain a minus sign?
It follows from how the horizontal component transforms under rotation and is confirmed by the cosine addition identity.
Can sin 75° be found exactly?
Yes. Using 45° + 30° gives (√6 + √2)/4.
Why does tan 75° involve a surd?
The exact ratio simplifies to 2 + √3 when the 45° and 30° values are combined.
Are the formulas valid in radians?
Yes. The identities hold for degree and radian representations of the same angles, provided units are interpreted consistently.
Why may a tangent addition expression be undefined?
Its denominator may be zero, corresponding to a direction where cosine vanishes and tangent itself is not defined.
Does the calculator give every interval solution?
No. An inverse trigonometric function generally returns a principal value; further permitted solutions must be determined from periodicity and the specified interval.
Are compound-angle formulae included in 2027 G3 A-Math?
Yes. The official K341 syllabus explicitly lists the sine, cosine and tangent addition and subtraction expansions under G1.
Continue through the Additional Mathematics series
For the closely related double-angle identities, see the rotating arrow that explains double-angle formulae.
For exact trigonometric ratios, read the triangle shadow that explains six trigonometric functions.
For principal inverse values and multiple angle solutions, read the solar panel hinge and inverse trigonometry.
The subject introduction is What Is Additional Mathematics?, with the wider How Additional Mathematics Works hub.
The immutable Clementi small-group Mathematics tutor reference illustrates why exact arithmetic, clear diagrams and targeted correction help students master later A-Math topics.
Official curriculum and model boundaries
The primary reference is SEAB’s 2027 G3 Additional Mathematics K341 syllabus (PDF), Geometry and Trigonometry G1, together with the official 2027 G3 subject listing.
The pointer and two turns are idealised plane geometry. They are not an actual navigation, mechanical or safety model.
The lasting lesson is that turning through a sum of angles combines the original horizontal and vertical components, and the compound-angle identities describe that combination exactly.