Summary (Grade 10 NSC Matric Mathematics): Revision Notes

📚 Revision Notes
Revision NotesModel AnswersQuizzesFlashcards

Summary

Trigonometric ratios for right-angled triangles

Trigonometry is built on three fundamental ratios that relate the sides of a right-angled triangle to its angles. These ratios are essential tools for solving geometric problems and are the foundation of all trigonometric calculations.

infoNote

Right-angled triangles provide the perfect foundation for understanding trigonometry because they have one angle of exactly 90°, making the relationships between sides and angles clear and predictable.

The three basic trigonometric ratios are:

  • Sine (sin) - relates the opposite side to the hypotenuse
  • Cosine (cos) - relates the adjacent side to the hypotenuse
  • Tangent (tan) - relates the opposite side to the adjacent side

Definitions of basic ratios

For any angle θ in a right-angled triangle:

  • sin⁡θ=oppositehypotenuse=yr\sin θ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}

  • cos⁡θ=adjacenthypotenuse=xr\cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}

  • tan⁡θ=oppositeadjacent=yx\tan θ = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}

chatImportant

Memory aid: Use SOH CAH TOA to remember these ratios:

  • Sine = Opposite over Hypotenuse
  • Cosine = Adjacent over Hypotenuse
  • Tangent = Opposite over Adjacent

Reciprocal trigonometric ratios

Each basic ratio has a reciprocal ratio that provides the inverse relationship:

  • Cosecant (cosec) - reciprocal of sine
  • Secant (sec) - reciprocal of cosine
  • Cotangent (cot) - reciprocal of tangent

Definitions of reciprocal ratios

  • cosec⁡θ=hypotenuseopposite=ry\cosec θ = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{r}{y}

  • sec⁡θ=hypotenuseadjacent=rx\sec θ = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{r}{x}

  • cot⁡θ=adjacentopposite=xy\cot θ = \frac{\text{adjacent}}{\text{opposite}} = \frac{x}{y}

infoNote

These reciprocal ratios are particularly useful when dealing with equations where the basic ratios appear in denominators, as they can help simplify complex expressions.

Solving trigonometric equations

You can combine algebraic principles with trigonometric ratios to solve equations involving unknown angles or sides. The key is to identify which ratio applies and then use algebraic manipulation to isolate the required value.

Working with special angles

For special angles (0°, 30°, 45°, 60°, and 90°), you can find exact trigonometric values without using a calculator. These values are essential for exam success and should be memorised.

lightbulbExample

Worked Example: Finding trigonometric values without a calculator

Problem: If tan⁡α=5−12\tan α = \frac{5}{-12} and 0° ≤ α ≤ 180°, find the value of 12cos⁡α\frac{12}{\cos α} without using a calculator.

Solution:

Step 1: Identify the quadrant Since tan α is negative and α is between 0° and 180°, angle α must be in the second quadrant (where tan is negative).

Step 2: Use the relationship between tan and the sides If tan⁡α=5−12\tan α = \frac{5}{-12}, we can think of a right triangle where:

  • Opposite side = 5
  • Adjacent side = -12

Step 3: Find the hypotenuse using Pythagoras

  • r2=52+(−12)2=25+144=169r^2 = 5^2 + (-12)^2 = 25 + 144 = 169
  • Therefore, r=13r = 13

Step 4: Find cos α

  • In the second quadrant, cos α is negative:
  • cos⁡α=adjacenthypotenuse=−1213\cos α = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-12}{13}

Step 5: Calculate the required value

  • 12cos⁡α=12−1213=12×13−12=−13\frac{12}{\cos α} = \frac{12}{\frac{-12}{13}} = 12 × \frac{13}{-12} = -13

Therefore, 12cos⁡α=−13\frac{12}{\cos α} = -13

Extended definitions

The trigonometric ratios can be extended beyond right-angled triangles to work with any angle. This extension allows trigonometry to be applied to circular functions and more complex geometric problems.

infoNote

This extension is crucial for understanding trigonometry in coordinate systems and for solving problems involving angles greater than 90°. It forms the foundation for advanced topics like periodic functions and wave analysis.

bookmarkSummary

Key Points to Remember:

  • SOH CAH TOA helps you remember the basic trigonometric ratios
  • Reciprocal ratios (cosec, sec, cot) are the inverses of the basic ratios
  • Special angles have exact values you can find without a calculator
  • Always consider the quadrant when determining signs of trigonometric values
  • Trigonometric ratios can be extended to any angle, not just acute angles in right triangles

Explore Grade 10 NSC Matric Mathematics Model Answers by Topics

Trigonometry

Euclidean Geometry

Analytical Geometry

Measurements

Statistics

Probability

Explore Grade 10 NSC Matric Mathematics Quizzes by Topics

Trigonometry

Euclidean Geometry

Analytical Geometry

Measurements

Statistics

Probability

Explore Grade 10 NSC Matric Mathematics Flashcards by Topics

Trigonometry

Euclidean Geometry

Analytical Geometry

Measurements

Statistics

Probability

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

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