'Trigonometry'

Angle Measurement

An angle is formed when a ray is rotated about its endpoint from initial to terminal position.

Degree and Radian Measure

• One radian: angle subtended at the centre by an arc equal in length to the radius.
• pi radians = 180 degrees
• 1 radian = 180/pi degrees
• 1 degree = pi/180 radian

Trigonometric Ratios

For an angle theta in standard position with point P(x, y) on terminal ray at distance r > 0:

• sin theta = y/r, cos theta = x/r, tan theta = y/x (where x != 0)
• csc theta = r/y, sec theta = r/x, cot theta = x/y (where denominators are non-zero)

Signs in Different Quadrants

Aid: 'Add Sugar To Coffee' — All positive in I, sin in II, tan in III, cos in IV.

Domain and Range

Compound Angles

• 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
• tan(A + B) = (tan A + tan B)/(1 - tan A tan B)
• tan(A - B) = (tan A - tan B)/(1 + tan A tan B)

Multiple and Submultiple Angles

• sin 2A = 2 sin A cos A = 2tan A/(1 + tan^2 A)
• cos 2A = cos^2 A - sin^2 A = 2cos^2 A - 1 = 1 - 2sin^2 A = (1 - tan^2 A)/(1 + tan^2 A)
• tan 2A = 2 tan A/(1 - tan^2 A)
• sin 3A = 3 sin A - 4 sin^3 A
• cos 3A = 4 cos^3 A - 3 cos A
• tan 3A = (3 tan A - tan^3 A)/(1 - 3 tan^2 A)

Transformation Formulas

Sum to Product:

• sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2)
• sin C - sin D = 2 cos((C+D)/2) sin((C-D)/2)
• cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2)
• cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2)

Product to Sum:

• 2 sin A cos B = sin(A+B) + sin(A-B)
• 2 cos A sin B = sin(A+B) - sin(A-B)
• 2 cos A cos B = cos(A+B) + cos(A-B)
• 2 sin A sin B = cos(A-B) - cos(A+B)

Sine Rule and Cosine Rule

For any triangle ABC with sides a, b, c opposite angles A, B, C:

Sine Rule: a/sin A = b/sin B = c/sin C = 2R (where R is circumradius)

Cosine Rule:

• a^2 = b^2 + c^2 - 2bc cos A
• b^2 = a^2 + c^2 - 2ac cos B
• c^2 = a^2 + b^2 - 2ab cos C

Trigonometric Equations

General solutions:

• If sin theta = sin alpha, then theta = npi + (-1)^n alpha
• If cos theta = cos alpha, then theta = 2npi pm alpha
• If tan theta = tan alpha, then theta = npi + alpha
• Where n in Z

Worked Examples

Example 1: Express pi/6 radians in degrees and 150 degrees in radians. Solution: pi/6 rad = 180/6 = 30 degrees. 150 degrees = 150 * pi/180 = 5pi/6 rad.

Example 2: Find the general solution of sin theta = 1/2. Solution: sin theta = sin(pi/6). General solution: theta = npi + (-1)^n pi/6, n in Z.

Example 3: Prove that sin 75 degrees = (sqrt(6) + sqrt(2))/4. Solution: sin 75 = sin(45 + 30) = sin45 cos30 + cos45 sin30 = (1/sqrt2)(sqrt3/2) + (1/sqrt2)(1/2) = (sqrt3 + 1)/(2sqrt2) = (sqrt6 + sqrt2)/4.

Common Mistakes

1. Ignoring quadrants: Always check the quadrant when finding angle values.
2. Confusing radian and degree mode: Ensure calculator is in correct mode.
3. Missing general solutions: Do not forget n in Z in trigonometric equations.
4. Sign errors: cos(A-B) = cos A cos B + sin A sin B (plus sign, not minus).

ISC Exam Focus

• Theory (70%): Proofs of identities, compound angle formulas, general solutions.
• Application (30%): Numerical problems using sine/cosine rule, transformation formulas.
• Typically 6-mark questions involving multiple angle formulas and equations.
• ISC frequently asks: "Find the general solution of ..." and "Prove that ...".

Self-Test Questions

Q1: Convert 210 degrees to radians. Answer: 210 * pi/180 = 7pi/6 radians.

Q2: If sin theta = 3/5 and theta is in quadrant II, find cos theta and tan theta. Answer: cos theta = -4/5 (negative in QII), tan theta = -3/4.

Q3: Prove that (sin 3A + sin A)/(cos 3A + cos A) = tan 2A. Answer: Using sum-to-product formulas, numerator = 2 sin2A cosA, denominator = 2 cos2A cosA, ratio = tan 2A.

Q4: Find the general solution of tan theta = sqrt(3). Answer: tan theta = tan(pi/3). General solution: theta = npi + pi/3, n in Z.

Q5: In triangle ABC, if a = 10, b = 12, and angle C = 60 degrees, find c. Answer: Using cosine rule, c^2 = 100 + 144 - 240cos60 = 244 - 120 = 124, so c = 2sqrt(31).

Q6: Prove that cos 2x = (1 - tan^2 x)/(1 + tan^2 x). Answer: cos 2x = (cos^2 x - sin^2 x)/(cos^2 x + sin^2 x) = (1 - tan^2 x)/(1 + tan^2 x).