Summary (Grade 11 NSC Matric Mathematics): Revision Notes

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Summary

Fundamental trigonometric identities

The square identity and quotient identity form the foundation of trigonometry. These relationships are always true, regardless of the angle size.

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The Square Identity (Pythagorean Identity)

cos⁡2θ+sin⁡2θ=1\cos^2 θ + \sin^2 θ = 1

This means that for any angle θ, when you square the cosine and sine values and add them together, you always get 1. This identity is the most fundamental relationship in trigonometry.

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The Quotient Identity

tan⁡θ=sin⁡θcos⁡θ\tan θ = \frac{\sin θ}{\cos θ}

This tells us that tangent is simply the ratio of sine to cosine for any angle. This relationship is essential for solving trigonometric equations.

Triangle notation and labelling

When working with triangles, we use a standard labelling system that helps us apply trigonometric rules correctly.

Figure

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Triangle Labelling Convention

In any triangle, we label the vertices with capital letters (A, B, C) and the sides opposite these vertices with corresponding lowercase letters (a, b, c).

  • Side 'a' is opposite angle A
  • Side 'b' is opposite angle B
  • Side 'c' is opposite angle C

This consistent labelling system is crucial for applying the sine and cosine rules correctly.

Special angle properties

Understanding how trigonometric functions behave with different angle transformations is crucial for solving complex problems.

Negative angles

  • sin⁡(−θ)=−sin⁡θ\sin(-θ) = -\sin θ (sine is an odd function)
  • cos⁡(−θ)=cos⁡θ\cos(-θ) = \cos θ (cosine is an even function)

Periodicity identities

Trigonometric functions repeat their values in regular intervals:

  • sin⁡(θ±360°)=sin⁡θ\sin(θ ± 360°) = \sin θ
  • cos⁡(θ±360°)=cos⁡θ\cos(θ ± 360°) = \cos θ

Co-function identities

These show the relationship between trigonometric functions of complementary angles:

  • sin⁡(90°−θ)=cos⁡θ\sin(90° - θ) = \cos θ
  • cos⁡(90°−θ)=sin⁡θ\cos(90° - θ) = \sin θ

Triangle solution rules

These three rules allow you to solve any triangle problem, depending on what information you're given.

Sine rule

Use when you have two angles and a side, or two sides and a non-included angle:

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Sine Rule

sin⁡Aa=sin⁡Bb=sin⁡Cc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Or in its reciprocal form: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Area rule

Use when you need to find the area of a triangle with two sides and the included angle:

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Area Rule Applications

  • Area of △ABC = 12bcsin⁡A\frac{1}{2}bc \sin A
  • Area of △ABC = 12acsin⁡B\frac{1}{2}ac \sin B
  • Area of △ABC = 12absin⁡C\frac{1}{2}ab \sin C

The key is having two sides and the included angle between them.

Cosine rule

Use when you have two sides and the included angle, or when you have all three sides:

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Cosine Rule

  • a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc \cos A
  • b2=a2+c2−2accos⁡Bb^2 = a^2 + c^2 - 2ac \cos B
  • c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

Note that when the angle is 90°, this reduces to Pythagoras' theorem since cos⁡90°=0\cos 90° = 0.

Quadrant signs (ASTC method)

The signs of trigonometric functions change depending on which quadrant the angle falls in.

Figure

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ASTC Method - Quadrant Signs

  • Quadrant I (0° to 90°): All functions are positive
  • Quadrant II (90° to 180°): Sine is positive, cosine and tangent are negative
  • Quadrant III (180° to 270°): Tangent is positive, sine and cosine are negative
  • Quadrant IV (270° to 360°): Cosine is positive, sine and tangent are negative

Remember the mnemonic: "All Students Take Calculus" or "All Stations To Central"

General solutions for trigonometric equations

When solving trigonometric equations, there are usually multiple solutions due to the periodic nature of these functions.

For sine equations

If sin⁡θ=x\sin θ = x:

  • θ=sin⁡−1x+k⋅360°θ = \sin^{-1} x + k \cdot 360°
  • or θ=(180°−sin⁡−1x)+k⋅360°θ = (180° - \sin^{-1} x) + k \cdot 360°

For cosine equations

If cos⁡θ=x\cos θ = x:

  • θ=cos⁡−1x+k⋅360°θ = \cos^{-1} x + k \cdot 360°
  • or θ=(360°−cos⁡−1x)+k⋅360°θ = (360° - \cos^{-1} x) + k \cdot 360°

For tangent equations

If tan⁡θ=x\tan θ = x:

  • θ=tan⁡−1x+k⋅180°θ = \tan^{-1} x + k \cdot 180°
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Understanding General Solutions

Where k∈Zk \in \mathbb{Z} (k is any integer)

The different forms account for the symmetry properties of trigonometric functions and ensure you find all possible solutions within any given domain.

Choosing the right rule

Knowing which rule to apply depends on the information given in the problem:

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Rule Selection Guide

Use the area rule when:

  • No perpendicular height is given
  • You have two sides and the included angle

Use the sine rule when:

  • No right angle is present
  • Two sides and a non-included angle are given
  • Two angles and one side are given

Use the cosine rule when:

  • No right angle is present
  • Two sides and the included angle are given
  • All three sides are given

Exam tips

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Critical Exam Strategies

  • Always check which quadrant your angle is in to determine the correct sign
  • When finding general solutions, consider the domain specified in the question
  • Use the appropriate inverse function on your calculator, but remember to find all solutions in the given interval
  • Draw diagrams when solving triangle problems to visualise the given information
  • Remember that the cosine rule reduces to Pythagoras' theorem when the angle is 90°
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Key Points to Remember

  • The fundamental identities cos⁡2θ+sin⁡2θ=1\cos^2 θ + \sin^2 θ = 1 and tan⁡θ=sin⁡θcos⁡θ\tan θ = \frac{\sin θ}{\cos θ} are your mathematical toolkit
  • ASTC helps you remember which trig functions are positive in each quadrant
  • Choose sine rule for non-included angles, cosine rule for included angles or three sides
  • General solutions account for the periodic nature of trigonometric functions
  • Always consider the domain when finding solutions to trigonometric equations

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