Further Trigonometry
FoundationModule 2 takes trigonometry further: angles in radians, three new reciprocal functions, and compound and double angle formulas for proving identities, simplifying expressions and solving trigonometric equations — the groundwork for calculus.
- Radians: 180° = π and 90° = π/2 — remember just this one conversion
- cosec x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x; combine with the square identities to simplify
- Compound angle formulas expand sums and differences; sin(A + B) is never sin A + sin B
文體小知識：數學概念說明文——由弧度法與三個倒數函數說起，重點講解複角公式與二倍角公式，並示範恆等式證明與在零至三百六十度區間解三角方程。
§Text Passage
1The compulsory course measures angles in degrees, with a right angle at ninety degrees; the extended course introduces radians: an angle whose arc length equals the radius is one radian. A semicircular arc has length π times the radius, so one hundred eighty degrees equals π radians. Radians make the formulas of trigonometry cleaner and are the standard unit for angle work in Module 2.
2Besides sine, cosine and tangent, this lesson adds three reciprocal functions: cosecant is the reciprocal of sine, secant the reciprocal of cosine, and cotangent the reciprocal of tangent. For example, if the sine of thirty degrees is one half, its cosecant is two. Together with the identities one plus tangent-squared equals secant-squared, and one plus cotangent-squared equals cosecant-squared, they open a much wider path for simplifying trigonometric expressions.
3Compound angle formulas handle functions of sums and differences of two angles: for example, the sine of a sum equals the sine of the first angle times the cosine of the second, plus the cosine of the first times the sine of the second. For instance, with forty-five plus thirty degrees, the formula gives root six plus root two all over four — exactly the sine of seventy-five degrees; unfamiliar angles are thus split into familiar ones.
4Double angle formulas are the special case where the two angles are equal: the sine of a double angle is twice the product of sine and cosine; the cosine of a double angle has three forms, commonly one minus twice sine-squared, or twice cosine-squared minus one. For example, if the sine of an angle is three fifths, the sine of its double angle is two times three fifths times four fifths, i.e. twenty-four twenty-fifths — a calculation used constantly in proofs and applications.
5These formulas also power equation solving and identity proving. For example, to solve the equation sine of double angle equals cosine, first write the left side as twice sine times cosine, move everything to one side and take out the common factor — two simple equations result, to be solved over the interval from zero to three hundred sixty degrees. Such questions appear regularly in the extended papers; the keys are knowing the formulas and listing every solution by quadrant, omitting none.
1Summary 重點摘要
Module 2 takes trigonometry further: angles in radians, three new reciprocal functions, and compound and double angle formulas for proving identities, simplifying expressions and solving trigonometric equations — the groundwork for calculus.
2Vocabulary cards Vocabulary
The angle subtending an arc equal in length to the radius; 180° = π radians, the standard angle unit in the extended course.
The reciprocal of sine, written cosec x = 1/sin x.
The reciprocal of cosine, written sec x = 1/cos x.
The reciprocal of tangent, written cot x = 1/tan x = cos x/sin x.
Formulas expanding trigonometric functions of a sum or difference of two angles, e.g. sin(A + B) = sin A cos B + cos A sin B.
The special case of the compound formulas with equal angles: sin 2A = 2 sin A cos A; cos 2A has three forms.
An equality holding for all admissible angles, marked with the triple bar ≡; a proof must transform one side into the other.
An equation containing trigonometric functions; within a given interval it usually has more than one solution, all to be listed by quadrant.
One of the four regions cut out by the axes; the quadrant decides the signs of the trigonometric values.
The interval length over which a function repeats; the periods of sine and cosine are 360° and of tangent 180°.
3Step by step Steps
- Step 1: Defining the radianRadians measure angles by arc length: in a circle of radius r, the angle subtending an arc of length r is 1 radian. The circumference is 2πr, so 360° = 2π radians, 180° = π radians and 90° = π/2. For example, to convert 60° to radians, multiply 60 by π and divide by 180, giving π/3; conversely, 5π/6 in degrees is 5π/6 × 180/π = 150°. Only one conversion is worth memorising: 180° corresponds to π. In later differentiation and integration all trigonometric formulas assume radians — build the habit now and it pays off.
- Step 2: The three reciprocal functionscosec x = 1/sin x, sec x = 1/cos x and cot x = 1/tan x (also cos x/sin x). For example, at x = 30°, sin x = 12 so cosec 30° = 2; and since cos 60° = 12, sec 60° = 2. Note that a reciprocal function is undefined where the original function is zero: sin 180° = 0, so cosec 180° does not exist. Together with the square identities 1 + tan²x = sec²x and 1 + cot²x = cosec²x, expressions simplify quickly — for instance sec²x − tan²x can be written down as 1 immediately.
- Step 3: Compound angle formulas for sine and cosineThe compound angle formulas expand functions of sums and differences: sin(A + B) = sin A cos B + cos A sin B; sin(A − B) = sin A cos B − cos A sin B; cos(A + B) = cos A cos B − sin A sin B; cos(A − B) = cos A cos B + sin A sin B. For example, to find sin 75°, read 75° as 45° + 30° and substitute, giving (√6 + √2)/4. Beware: sin(A + B) is not sin A + sin B — the most common error. Verify with A = B = 30°: the left side sin 60° is about 0.866 while the right side is 1; the formula is therefore essential.
- Step 4: The compound formula for tangentFor tangent, tan(A + B) = (tan A + tan B)/(1 − tan A tan B); for the difference the numerator becomes tan A − tan B and the denominator 1 + tan A tan B. For example, tan(45° + 30°) = (1 + 1/√3)/(1 − 1/√3), which simplifies to 2 + √3 — precisely tan 75°. The formula fails when tan A tan B = 1 because the denominator vanishes; this corresponds to A + B = 90°, where the tangent of the sum is undefined anyway, so the failure matches reality. Memory aid: the numerator adds or subtracts the two tangents, the denominator is 1 minus or plus the product — the signs cross over.
- Step 5: The double angle formulasSetting B = A in the compound formulas gives: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1, with the three forms chosen as the question demands; and tan 2A = 2 tan A/(1 − tan²A). For example, given sin A = 35 with A acute, cos A = 45, so sin 2A = 2 × (35) × (45) = 2425. The three cosine forms swap freely in identity proofs: whenever sin²A appears replace it by (1 − cos 2A)/2, and whenever cos²A appears replace it by (1 + cos 2A)/2 — quadratic terms immediately drop to first power.
- Step 6: Proving identities with the formulasThe general strategy: start from the more complicated side, convert everything to sines and cosines, and apply the formulas step by step towards the other side, citing the formula used on every line. For example, to prove sin 2A = 2 sin A cos A, simply put B = A in sin(A + B) — one line; to prove sin(A + B) sin(A − B) = sin²A − sin²B, expand the left side and collect like terms. Discipline: keep the equal signs moving in one direction only — never transform both sides simultaneously to meet in the middle, and never work backwards; every step must be traceable, which is exactly what markers check.
- Step 7: Solving trigonometric equations from 0° to 360°First reduce the equation to a simple one involving a single function, then list all solutions using the period and the quadrants. For example, solve sin 2x = cos x for 0° ≤ x ≤ 360°. Write the left side as 2 sin x cos x by the double angle formula; moving everything left gives cos x(2 sin x − 1) = 0, so cos x = 0 or sin x = 12. The first gives x = 90°, 270°; the second, positive in quadrants I and II, gives x = 30°, 150°. For equations like sin 2x = 1, put θ = 2x so the range doubles to 0°–720°, solve θ = 90°, 450°, then divide by 2 to get x = 45°, 225° — when the angle is doubled the range doubles too, the classic place to lose solutions.
5Worked example Example
Example: (a) prove the identity sin(A + B) − sin(A − B) = 2 cos A sin B; (b) solve sin 2x = sin x for 0° ≤ x ≤ 360°.
- (a) Expand the left side: sin(A + B) − sin(A − B) = (sin A cos B + cos A sin B) − (sin A cos B − cos A sin B)
- Collect terms: the two sin A cos B terms cancel, leaving cos A sin B + cos A sin B = 2 cos A sin B, as required
- (b) Apply the double angle formula: sin 2x = 2 sin x cos x, so the equation becomes 2 sin x cos x = sin x
- Move everything left and factorise: 2 sin x cos x − sin x = 0, i.e. sin x(2 cos x − 1) = 0
- Solving sin x = 0 gives x = 0°, 180°, 360°; solving cos x = 12 gives x = 60°, 300°
- Answer: (a) Expanding by the compound angle formulas and collecting terms, both sides are equal, proving the identity. (b) x = 0°, 60°, 180°, 300°, 360°. Prove identities by simplifying the more complicated side; for equations, reduce the degree or factorise first, and remember sin x = 0 has three solutions in the interval — do not omit 0° and 360°.
6Common mistakes Mistakes
7Quick check Quick check
Q1How is the radian defined, and how many radians is 180°?
Q2Use the compound angle formula to find sin 75°.
Q3Given sin A = 35 (A acute), find sin 2A.
Q4Solve cos x = −12 for 0° ≤ x ≤ 360°.
Q5What is the usual strategy for proving an identity?
8Recap Recap
- Radians: 180° = π and 90° = π/2 — remember just this one conversion
- cosec x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x; combine with the square identities to simplify
- Compound angle formulas expand sums and differences; sin(A + B) is never sin A + sin B
- Double angle formulas come from putting B = A; cos 2A has three interchangeable forms
- Reduce equations to one function, list all solutions by quadrant; when the angle doubles, the range doubles too