13.2 Trigonometry

13.2 Trigonometry for the PAF GD Pilot Initial Test

Quick Answer: PAF trigonometry questions test three things: radian-degree conversion (pi radians = 180 degrees), the six trig ratios and their identities (sin^2 + cos^2 = 1), and the period formula 2*pi/|b| for sinusoidal functions. You will also see the sine rule and cosine rule for non-right triangles. Exact values at 0, 30, 45, 60, 90 degrees are expected knowledge.

Radian and Degree Conversion

Angles are measured in degrees or radians. The bridge between them is the identity:

pi radians = 180 degrees

From this:

• 1 radian = 180/pi degrees, approx 57.3 degrees.
• 1 degree = pi/180 radians, approx 0.01745 radians.

Common conversions: 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, 180 degrees = pi, 360 degrees = 2*pi.

The Six Trig Ratios

For a right triangle with angle theta:

• sin(theta) = opposite / hypotenuse
• cos(theta) = adjacent / hypotenuse
• tan(theta) = opposite / adjacent = sin(theta)/cos(theta)
• cosec(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), cot(theta) = 1/tan(theta).

Pythagorean and Compound-Angle Identities

The Pythagorean identity is the cornerstone:

sin^2(theta) + cos^2(theta) = 1

Dividing by cos^2 gives 1 + tan^2(theta) = sec^2(theta); dividing by sin^2 gives cot^2(theta) + 1 = cosec^2(theta).

Compound-angle formulas (examiners love these):

• 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)

The double-angle identities follow by setting B = A: sin(2A) = 2 sin A cos A, cos(2A) = cos^2 A - sin^2 A = 2 cos^2 A - 1 = 1 - 2 sin^2 A.

Sine and Cosine Rules

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. Use when you know a side and its opposite angle (the AAS, ASA, or SSA cases).
• Cosine rule: c^2 = a^2 + b^2 - 2ab cos C. Use for SAS (two sides and the included angle) or SSS (three sides, find an angle).

The cosine rule generalises Pythagoras: when C = 90 degrees, cos C = 0 and it reduces to c^2 = a^2 + b^2.

Periodicity of Sinusoidal Functions

A function of the form y = asin(bx + c) + d or y = acos(bx + c) + d has:

• Amplitude = |a|.
• Period = 2*pi/|b|.
• Phase shift = -c/b.
• Vertical shift = d.

Worked example. Find the period of y = 3*cos(2x).

• Here b = 2, so period = 2pi/|b| = 2pi/2 = pi.
• The amplitude is |3| = 3, but the question only asks for the period.

This exact question (period of 3cos(2x) = pi) is a known PAF Initial Test item.

Exact Trig Values Table

A useful mnemonic for sin: as the angle goes 0, 30, 45, 60, 90, the values are sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2, sqrt(4)/2, i.e. 0, 1/2, sqrt(2)/2, sqrt(3)/2, 1. Cosine is the same list reversed.

Sine Rule Worked Example

In triangle ABC, angle A = 40 degrees, angle B = 60 degrees, and side a (opposite A) = 8 cm. Find side b (opposite B).

1. Angle C = 180 - 40 - 60 = 80 degrees (angles sum to 180).
2. Sine rule: a/sin A = b/sin B, so b = a * sin B / sin A = 8 * sin 60 / sin 40.
3. Using sin 60 = sqrt(3)/2 approx 0.866 and sin 40 approx 0.643, b approx 8 * 0.866 / 0.643 approx 10.77 cm.

The sine rule is the right tool when you have a side paired with its opposite angle. If instead you have two sides and the included angle (SAS), reach for the cosine rule; mixing the two is a common error.

Identity Proof (Exam-Style)

Prove that tan(theta) + cot(theta) = sec(theta)*cosec(theta).

1. Write everything in sines and cosines: tan = sin/cos, cot = cos/sin.
2. Left-hand side: sin(theta)/cos(theta) + cos(theta)/sin(theta) = [sin^2(theta) + cos^2(theta)] / [sin(theta)*cos(theta)].
3. Use the Pythagorean identity sin^2 + cos^2 = 1 in the numerator: = 1 / [sin(theta)*cos(theta)].
4. Split the fraction: = (1/cos(theta)) * (1/sin(theta)) = sec(theta)*cosec(theta).
5. This equals the right-hand side, so the identity is proved.

The strategy generalises: to prove a trig identity, convert all ratios to sin and cos, combine over a common denominator, then apply sin^2 + cos^2 = 1. Never cross-multiply or assume the identity to prove it — work one side into the other.

Common Traps

• Period vs amplitude: the period of asin(bx) is 2pi/|b|, not 2pi/|a|. The amplitude |a| controls height; the coefficient b controls the rate of oscillation. For 3*cos(2x), b = 2 gives period pi; a = 3 only sets the amplitude.
• Sign in cosine rule: c^2 = a^2 + b^2 - 2abcos(C) has a minus before the 2abcos(C) term. Dropping the minus (writing +2ab*cos C) is a frequent slip that produces the wrong side length for obtuse angles.
• Quadrant signs: sin is positive in quadrants I and II, cos in I and IV, tan in I and III (the ASTC rule: All, Sin, Tan, Cos). If sin(theta) = 3/5 and theta is in quadrant II, cos(theta) = -4/5, not +4/5.