Revision (Grade 12 NSC Matric Mathematics): Revision Notes

📚 Revision Notes
Revision NotesModel AnswersQuizzesFlashcards

Revision

Fundamental trigonometric ratios

Trigonometric ratios are mathematical relationships that connect the angles of a triangle to the lengths of its sides. These ratios form the foundation of trigonometry and are essential for solving problems involving angles and distances.

In a right-angled triangle, we define three primary trigonometric ratios using the relationship between sides relative to a specific angle.

For any angle in a right triangle:

  • Sine (sin) = opposite sidehypotenuse\frac{\text{opposite side}}{\text{hypotenuse}}
  • Cosine (cos) = adjacent sidehypotenuse\frac{\text{adjacent side}}{\text{hypotenuse}}
  • Tangent (tan) = opposite sideadjacent side\frac{\text{opposite side}}{\text{adjacent side}}
infoNote

These ratios remain constant for any given angle, regardless of the triangle's size. This fundamental property makes trigonometry a powerful tool for solving real-world problems involving angles and distances.

Coordinate plane trigonometry

When we extend trigonometric concepts to the coordinate plane, we can define these ratios for any angle, not just those in right triangles.

For a point P(x,y) at distance r from the origin, forming angle α with the positive x-axis:

  • sin⁡α=yr\sin α = \frac{y}{r}
  • cos⁡α=xr\cos α = \frac{x}{r}
  • tan⁡α=yx\tan α = \frac{y}{x}
infoNote

This coordinate approach allows us to work with angles greater than 90° and negative angles, extending trigonometry beyond the limitations of right triangles.

Signs of trigonometric functions - CAST diagram

The CAST diagram shows which trigonometric functions are positive in each quadrant of the coordinate plane. This is crucial for determining the correct signs when working with angles in different quadrants.

Figure

chatImportant

Memory aid: "All Students Take Calculus"

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

The sign of a trigonometric function depends on the signs of x and y coordinates in each quadrant.

Reduction formulae and co-functions

Reduction formulae allow us to express trigonometric functions of any angle in terms of an acute angle (between 0° and 90°). This is essential for simplifying complex trigonometric expressions.

Key reduction rules:

First quadrant (0° to 90°):

  • All functions remain positive
  • sin⁡(90°−θ)=cos⁡θ\sin(90° - θ) = \cos θ
  • cos⁡(90°−θ)=sin⁡θ\cos(90° - θ) = \sin θ

Second quadrant (90° to 180°):

  • Only sine is positive
  • sin⁡(180°−θ)=sin⁡θ\sin(180° - θ) = \sin θ
  • cos⁡(180°−θ)=−cos⁡θ\cos(180° - θ) = -\cos θ
  • tan⁡(180°−θ)=−tan⁡θ\tan(180° - θ) = -\tan θ

Third quadrant (180° to 270°):

  • Only tangent is positive
  • sin⁡(180°+θ)=−sin⁡θ\sin(180° + θ) = -\sin θ
  • cos⁡(180°+θ)=−cos⁡θ\cos(180° + θ) = -\cos θ
  • tan⁡(180°+θ)=tan⁡θ\tan(180° + θ) = \tan θ

Fourth quadrant (270° to 360°):

  • Only cosine is positive
  • sin⁡(360°−θ)=−sin⁡θ\sin(360° - θ) = -\sin θ
  • cos⁡(360°−θ)=cos⁡θ\cos(360° - θ) = \cos θ
  • tan⁡(360°−θ)=−tan⁡θ\tan(360° - θ) = -\tan θ

Co-function relationships:

Co-functions are complementary trigonometric functions that relate angles summing to 90°:

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

These relationships show the complementary nature of sine and cosine functions.

Negative angles

Trigonometric functions of negative angles follow specific patterns that are important to remember:

  • 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)
  • tan⁡(−θ)=−tan⁡θ\tan(-θ) = -\tan θ (tangent is an odd function)

These relationships help us evaluate trigonometric functions for negative angle measures.

Special angle triangles

Special triangles provide exact values for trigonometric functions at commonly used angles. These values are essential for solving problems without a calculator.

The 30°-60°-90° triangle has sides in the ratio 1:3:21 : \sqrt{3} : 2 The 45°-45°-90° triangle has sides in the ratio 1:1:21 : 1 : \sqrt{2}

chatImportant

Special angle values must be memorised:

  • 0°: cos⁡=1\cos = 1, sin⁡=0\sin = 0, tan⁡=0\tan = 0
  • 30°: cos⁡=32\cos = \frac{\sqrt{3}}{2}, sin⁡=12\sin = \frac{1}{2}, tan⁡=13\tan = \frac{1}{\sqrt{3}}
  • 45°: cos⁡=12\cos = \frac{1}{\sqrt{2}}, sin⁡=12\sin = \frac{1}{\sqrt{2}}, tan⁡=1\tan = 1
  • 60°: cos⁡=12\cos = \frac{1}{2}, sin⁡=32\sin = \frac{\sqrt{3}}{2}, tan⁡=3\tan = \sqrt{3}
  • 90°: cos⁡=0\cos = 0, sin⁡=1\sin = 1, tan⁡=\tan = undefined

These exact values are fundamental for evaluating trigonometric expressions accurately.

Fundamental trigonometric identities

Trigonometric identities are equations that are true for all valid values of the variable. They are powerful tools for simplifying expressions and solving equations.

Quotient identity:

tan⁡θ=sin⁡θcos⁡θ (where cos⁡θ≠0)\tan θ = \frac{\sin θ}{\cos θ} \text{ (where } \cos θ ≠ 0\text{)}

This shows the relationship between the three basic trigonometric functions.

Pythagorean identity:

sin⁡2θ+cos⁡2θ=1\sin² θ + \cos² θ = 1

infoNote

This fundamental identity leads to several useful variations:

  • sin⁡2θ=1−cos⁡2θ\sin² θ = 1 - \cos² θ
  • cos⁡2θ=1−sin⁡2θ\cos² θ = 1 - \sin² θ
  • sin⁡θ=±1−cos⁡2θ\sin θ = ±\sqrt{1 - \cos² θ}
  • cos⁡θ=±1−sin⁡2θ\cos θ = ±\sqrt{1 - \sin² θ}

The ± sign depends on the quadrant in which the angle lies.

Worked examples

lightbulbExample

Worked Example 1: Evaluating trigonometric expressions

Question: Determine the value of cos⁡420°−sin⁡225°cos⁡(−45°)/tan⁡315°\cos 420° - \sin 225° \cos(-45°) / \tan 315°

Step 1: Use reduction formulae to express each term using acute angles

  • cos⁡420°=cos⁡(360°+60°)=cos⁡60°\cos 420° = \cos(360° + 60°) = \cos 60°
  • sin⁡225°=sin⁡(180°+45°)=−sin⁡45°\sin 225° = \sin(180° + 45°) = -\sin 45°
  • cos⁡(−45°)=cos⁡45°\cos(-45°) = \cos 45°
  • tan⁡315°=tan⁡(360°−45°)=−tan⁡45°\tan 315° = \tan(360° - 45°) = -\tan 45°

Step 2: Substitute special angle values

  • cos⁡60°=12\cos 60° = \frac{1}{2}
  • sin⁡45°=12\sin 45° = \frac{1}{\sqrt{2}}, so −sin⁡45°=−12-\sin 45° = -\frac{1}{\sqrt{2}}
  • cos⁡45°=12\cos 45° = \frac{1}{\sqrt{2}}
  • tan⁡45°=1\tan 45° = 1, so −tan⁡45°=−1-\tan 45° = -1

Step 3: Calculate the expression

=12+(−12)(12)−1= \frac{1}{2} + \frac{\left(-\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)}{-1}

=12+−12−1=12+12=1= \frac{1}{2} + \frac{-\frac{1}{2}}{-1} = \frac{1}{2} + \frac{1}{2} = 1

lightbulbExample

Worked Example 2: Proving trigonometric identities

Question: Prove that sin⁡2α−(tan⁡α−cos⁡α)(tan⁡α+cos⁡α)=cos⁡2α−sin⁡2αcos⁡2α\sin² α - (\tan α - \cos α)(\tan α + \cos α) = \frac{\cos² α - \sin² α}{\cos² α}

Step 1: Simplify the left-hand side

  • LHS=sin⁡2α−(tan⁡2α−cos⁡2α)\text{LHS} = \sin² α - (\tan² α - \cos² α)
  • =sin⁡2α−tan⁡2α+cos⁡2α= \sin² α - \tan² α + \cos² α
  • =(sin⁡2α+cos⁡2α)−tan⁡2α= (\sin² α + \cos² α) - \tan² α
  • =1−tan⁡2α= 1 - \tan² α

Step 2: Express in terms of sine and cosine

  • =1−sin⁡2αcos⁡2α=cos⁡2α−sin⁡2αcos⁡2α= 1 - \frac{\sin² α}{\cos² α} = \frac{\cos² α - \sin² α}{\cos² α}

Step 3: This equals the right-hand side, so the identity is proven.

Restrictions: The identity is undefined when cos⁡α=0\cos α = 0, which occurs at α=90°α = 90° and α=270°α = 270°.

lightbulbExample

Worked Example 3: Using reduction formulae

Question: Simplify sin⁡150°+cos⁡240°−tan⁡315°\sin 150° + \cos 240° - \tan 315°

Step 1: Apply reduction formulae

  • sin⁡150°=sin⁡(180°−30°)=sin⁡30°=12\sin 150° = \sin(180° - 30°) = \sin 30° = \frac{1}{2}
  • cos⁡240°=cos⁡(180°+60°)=−cos⁡60°=−12\cos 240° = \cos(180° + 60°) = -\cos 60° = -\frac{1}{2}
  • tan⁡315°=tan⁡(360°−45°)=−tan⁡45°=−1\tan 315° = \tan(360° - 45°) = -\tan 45° = -1

Step 2: Calculate

  • =12+(−12)−(−1)=12−12+1=1= \frac{1}{2} + \left(-\frac{1}{2}\right) - (-1) = \frac{1}{2} - \frac{1}{2} + 1 = 1

Exam tips and common traps

Understanding these key strategies will help you avoid common pitfalls and approach trigonometry problems with confidence:

  • Always check which quadrant your angle is in to determine the correct sign
  • Remember that tan⁡θ\tan θ is undefined when cos⁡θ=0\cos θ = 0 (at 90° and 270°)
  • When proving identities, work with one side at a time - never cross-multiply
  • Always state restrictions where trigonometric functions are undefined
  • Use exact values from special triangles rather than decimal approximations
chatImportant

Common mistake to avoid: Writing trigonometric ratios without angles (e.g., tan⁡=sin⁡/cos⁡\tan = \sin/\cos has no meaning - you must specify the angle)

Memory aid: Use CAST ("All Students Take Calculus") for quadrant signs

bookmarkSummary

Key Points to Remember:

  • Trigonometric ratios are defined as opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent in right triangles
  • CAST diagram determines which functions are positive: All (I), Sine (II), Tangent (III), Cosine (IV)
  • Reduction formulae convert any angle to an equivalent acute angle expression
  • Special angles (30°, 45°, 60°) have exact trigonometric values that must be memorised
  • Fundamental identity: sin⁡2θ+cos⁡2θ=1\sin² θ + \cos² θ = 1 is the foundation for many algebraic manipulations

Explore Grade 12 NSC Matric Mathematics Model Answers by Topics

Trigonometry

Analytical Geometry

Euclidean Geometry

Statistics

Probability

Explore Grade 12 NSC Matric Mathematics Quizzes by Topics

Trigonometry

Analytical Geometry

Euclidean Geometry

Statistics

Probability

Explore Grade 12 NSC Matric Mathematics Flashcards by Topics

Trigonometry

Analytical Geometry

Euclidean Geometry

Statistics

Probability

Join 100,000+ NSC Matric students studying Revision Notes with us.

Select your subjects, and get access to A+ resources today.