Geometry, Trig & Calculus

Trigonometric Functions, Identities, Sine Rule and Cosine Rule

Grade 10 · Grade 11 · Grade 12

✓By the end of this lesson students will be able to define and apply the basic trigonometric ratios and their reciprocals.
✓By the end of this lesson students will be able to simplify trigonometric expressions and prove identities using fundamental identities and reduction formulae.
✓By the end of this lesson students will be able to apply compound and double angle identities to simplify expressions and solve equations.
✓By the end of this lesson students will be able to solve problems in two dimensions using the Sine Rule, Cosine Rule, and Area Rule.

Key concepts

Basic Trigonometric Ratios (SOH CAH TOA)

For a right-angled triangle, the basic trigonometric ratios relate the angles to the lengths of the sides. In the Cartesian plane, for an angle θ in standard position with a point P(x;y) on its terminal arm and r as the distance from the origin to P (where r = √(x²+y²)):

sin θ = opposite / hypotenuse = y/r\ncos θ = adjacent / hypotenuse = x/r\ntan θ = opposite / adjacent = y/x

Reciprocal Trigonometric Ratios

These are the reciprocals of the basic trigonometric ratios:

cosec θ = 1 / sin θ = r/y\nsec θ = 1 / cos θ = r/x\ncot θ = 1 / tan θ = x/y

Special Angles

The trigonometric ratios for 0°, 30°, 45°, 60°, and 90° can be determined without a calculator using special triangles or the Cartesian plane.

sin 0° = 0, cos 0° = 1, tan 0° = 0\nsin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3\nsin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1\nsin 60° = √3/2, cos 60° = 1/2, tan 60° = √3\nsin 90° = 1, cos 90° = 0, tan 90° = undefined

Reduction Formulae and CAST Diagram

Reduction formulae allow us to express trigonometric ratios of angles greater than 90° (or negative angles) in terms of acute angles. The CAST diagram helps determine the sign of the ratio in each quadrant.

Quadrant I (0° < θ < 90°): All ratios positive\nQuadrant II (90° < θ < 180°): Sin positive (180°-θ)\nQuadrant III (180° < θ < 270°): Tan positive (180°+θ)\nQuadrant IV (270° < θ < 360°): Cos positive (360°-θ or -θ)\n\nExamples:\nsin(180°-θ) = sin θ\ncos(180°+θ) = -cos θ\ntan(360°-θ) = -tan θ\nsin(-θ) = -sin θ\ncos(-θ) = cos θ\ntan(-θ) = -tan θ\nsin(90°-θ) = cos θ\ncos(90°-θ) = sin θ

Fundamental Identities

These identities are derived from the definitions of the trigonometric ratios and the Pythagorean theorem.

Quotient Identity: tan θ = sin θ / cos θ\nPythagorean Identity: sin²θ + cos²θ = 1

Compound Angle Identities (Grade 12)

These identities express the trigonometric ratios of the sum or difference of two angles.

cos(A + B) = cos A cos B - sin A sin B\ncos(A - B) = cos A cos B + sin A sin B\nsin(A + B) = sin A cos B + cos A sin B\nsin(A - B) = sin A cos B - cos A sin B

Double Angle Identities (Grade 12)

These are special cases of compound angle identities where the two angles are equal.

sin 2A = 2 sin A cos A\ncos 2A = cos²A - sin²A\ncos 2A = 2cos²A - 1\ncos 2A = 1 - 2sin²A

Sine Rule (for any triangle)

The Sine Rule relates the sides of a triangle to the sines of its opposite angles. It is used when you have a side and its opposite angle, and another side or angle.

a / sin A = b / sin B = c / sin C

Cosine Rule (for any triangle)

The Cosine Rule relates the sides and angles of a triangle. It is used when you have two sides and the included angle (SAS) to find the third side, or all three sides (SSS) to find an angle.

a² = b² + c² - 2bc cos A\nb² = a² + c² - 2ac cos B\nc² = a² + b² - 2ab cos C

Area Rule (for any triangle)

The Area Rule calculates the area of any triangle using two sides and the included angle.

Area = 1/2 ab sin C\nArea = 1/2 bc sin A\nArea = 1/2 ac sin B

Key facts to remember

1The CAST diagram helps determine the sign of trigonometric ratios in different quadrants.
2The fundamental identities are tan θ = sin θ / cos θ and sin²θ + cos²θ = 1.
3Reduction formulae allow you to express any angle in terms of an acute angle.
4The Sine Rule (a/sin A = b/sin B = c/sin C) is used when you have a side and its opposite angle, and another side or angle.
5The Cosine Rule (a² = b² + c² - 2bc cos A) is used when you have two sides and the included angle (SAS) or all three sides (SSS).
6The Area Rule (Area = 1/2 ab sin C) calculates the area of any triangle using two sides and the included angle.
7Compound and Double Angle identities are essential for Grade 12 trigonometry.

Worked examples

Example 1

Prove the identity: (1 - cos²x) / (sin x cos x) = tan x

ILHS = (1 - cos²x) / (sin x cos x)

IIUsing the Pythagorean Identity, sin²x + cos²x = 1, so 1 - cos²x = sin²x.

IIILHS = sin²x / (sin x cos x)

IVCancel out a sin x from the numerator and denominator.

VLHS = sin x / cos x

VIUsing the Quotient Identity, tan x = sin x / cos x.

VIILHS = tan x

VIIITherefore, LHS = RHS.

Answer

LHS = tan x = RHS

Always start from one side (LHS or RHS) and work towards the other. Do not work on both sides simultaneously.

Example 2

Simplify the following expression without using a calculator: (sin(180°-x)cos(360°-x)tan(-x)) / (cos(90°+x)sin(x-180°))

IApply reduction formulae and negative angle identities to each term:

IIsin(180°-x) = sin x (Quadrant II, sin is positive)

IIIcos(360°-x) = cos x (Quadrant IV, cos is positive)

IVtan(-x) = -tan x (tan is odd function)

Vcos(90°+x) = -sin x (Quadrant II, cos is negative, co-function)

VIsin(x-180°) = sin(-(180°-x)) = -sin(180°-x) = -sin x (sin is odd function, then reduction)

VIISubstitute these into the expression:

VIIIExpression = (sin x * cos x * (-tan x)) / (-sin x * (-sin x))

9Expression = (-sin x cos x tan x) / (sin²x)

10Replace tan x with sin x / cos x:

11Expression = (-sin x cos x (sin x / cos x)) / (sin²x)

12Cancel out cos x in the numerator:

13Expression = (-sin x * sin x) / (sin²x)

14Expression = -sin²x / sin²x

15Expression = -1

Answer

-1

Pay close attention to the signs in each quadrant and when applying co-functions.

Example 3

In triangle PQR, PQ = 8 cm, PR = 10 cm and angle QPR = 60°. Calculate the length of QR and the area of triangle PQR.

ITo find the length of QR, we have two sides and the included angle (SAS), so we use the Cosine Rule.

IILet p = QR, q = PR = 10 cm, r = PQ = 8 cm, and angle P = 60°.

IIIp² = q² + r² - 2qr cos P

IVp² = (10)² + (8)² - 2(10)(8) cos 60°

Vp² = 100 + 64 - 160(1/2)

VIp² = 164 - 80

VIIp² = 84

VIIIp = √84 = 2√21 cm

9To find the area of triangle PQR, we use the Area Rule.

10Area = 1/2 qr sin P

11Area = 1/2 (10)(8) sin 60°

12Area = 1/2 (80) (√3/2)

13Area = 40 (√3/2)

14Area = 20√3 cm²

Answer

QR = 2√21 cm, Area = 20√3 cm²

Remember to use the correct formula based on the given information (SAS for Cosine Rule, two sides and included angle for Area Rule).

Common mistakes

✗Incorrectly applying the signs of trigonometric ratios in different quadrants (CAST diagram errors).
✗Confusing `sin²θ` with `sin θ²` or `sin 2θ`.
✗Algebraic errors when manipulating trigonometric expressions or proving identities.
✗Not knowing when to use the Sine Rule versus the Cosine Rule.
✗Assuming a triangle is right-angled when it is not, and incorrectly applying SOH CAH TOA directly.

Exam tips

★Memorise all fundamental identities, reduction formulae, special angle values, and the Sine, Cosine, and Area Rules.
★Practice using the CAST diagram and reduction formulae until they are second nature.
★When proving identities, always work from one side (LHS or RHS) towards the other, showing all steps clearly.
★Draw clear, labelled diagrams for geometry problems involving the Sine and Cosine Rules to help visualise the problem and identify knowns/unknowns.
★Always check your calculator mode (degrees or radians) before performing calculations.

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