Trigonometry Study Notes

Compound Angles

Concept Explanation: Compound angles refer to the sum or difference of two angles, commonly represented as (A + B) or (A – B). The key formulas allow you to express trigonometric functions of compound angles in terms of the functions of their individual angles.

Important Facts:

• The main compound angle identities are:

Example: Calculate \sin(50^\circ + 40^\circ). Step 1: Write the formula.

\sin(A + B) = \sin A \cos B + \cos A \sin B

Step 2: Substitute the values.

\sin(50^\circ + 40^\circ) = \sin 50^\circ \cos 40^\circ + \cos 50^\circ \sin 40^\circ

Step 3: Simplify using a calculator.

= (0,7660 \times 0,7660) + (0,6428 \times 0,6428)

= 0,5868 + 0,4132 = 1

Final answer:

\sin(90^\circ) = 1

Common Mistake: Learners often use the wrong sign in the expanded formula. Remember: the sign within the formula for sine stays the same, for cosine it changes sign.

Exam Tip: Always write out the full formula before substituting the values.

Double Angle Identities

Concept Explanation: Double angle identities are used for trigonometric expressions involving twice an angle, such as 2A.

Important Facts:

• Main double angle formulas:

Example: Simplify 2\sin 25^\circ\cos 25^\circ. Step 1: Use the double angle identity for sine.

\sin 2A = 2 \sin A \cos A

So,

2\sin 25^\circ\cos 25^\circ = \sin(2 \times 25^\circ)

= \sin 50^\circ

Common Mistake: Forgetting to double the angle after applying the double angle formula, e.g., using 2\sin\theta\cos\theta = \sin\theta instead of \sin2\theta.

Exam Tip: Know all three forms of the cosine double angle formula.

Reduction Formulae

Concept Explanation: Reduction formulae enable the simplification of trigonometric expressions involving (180° ± x), (360° ± x), (90° ± x), etc.

Important Facts:

• Use the CAST diagram to determine the sign of the function in each quadrant.
• Key reduction identities include:

Example: Evaluate \cos(180^\circ - 40^\circ). Step 1: Write the formula.

\cos(180^\circ - x) = -\cos x

Step 2: Substitute values.

\cos(180^\circ - 40^\circ) = -\cos 40^\circ

= -0,7660

Common Mistake: Using the wrong sign for the quadrant.

Exam Tip: Always check the quadrant with a CAST diagram.

Trigonometric Identities

Concept Explanation: Trigonometric identities are equations true for all angles, used to simplify or prove trigonometric expressions.

Important Identities:

• \sin^2\theta + \cos^2\theta = 1
  
• 1 + \tan^2\theta = \sec^2\theta
  
• 1 + \cot^2\theta = \csc^2\theta
  

Example: Prove that 1 - \sin^2\theta = \cos^2\theta. Step 1: Start with the given expression.

1 - \sin^2\theta

Step 2: Use the Pythagorean identity.

\sin^2\theta + \cos^2\theta = 1

Rearrange:

\cos^2\theta = 1 - \sin^2\theta

Therefore, 1 - \sin^2\theta = \cos^2\theta.

Common Mistake: Not recognizing equivalent forms of identities.

Exam Tip: Show all steps when proving identities in tests.

Solving Trigonometric Equations

Concept Explanation: Solving trig equations means finding all angles that satisfy a trigonometric equation, often within a given interval.

Steps:

• Isolate the trig function first.
• Use reduction, factorisation or identities to simplify if needed.
• Find reference angle using calculator.
• Apply general solution based on function and interval.

Example: Solve \sin x = \frac{1}{2} for 0^\circ \leq x \leq 360^\circ. Step 1: Find the reference angle.

\sin^{-1}(\frac{1}{2}) = 30^\circ

Step 2: Write the general solution from the sine graph (since sine is positive in I and II quadrants).

x = 30^\circ \quad\text{or}\quad x = 180^\circ - 30^\circ

x = 30^\circ \text{ or } x = 150^\circ

Common Mistake: Forgetting all possible solutions in the range.

Exam Tip: Use the CAST diagram to determine where the function is positive or negative.

General Solutions

Concept Explanation: The general solution represents all possible solutions to a trig equation, using k as the integer constant.

General Solution Forms:

• For sine:

Example: Write the general solution for \tan x = 1. Reference angle:

x = 45^\circ

General solution:

x = 45^\circ + 180^\circ k

Common Mistake: Mixing up the periodicity of sine/cosine (360°) and tangent (180°).

Exam Tip: Always include k \in \mathbb{Z} in your answer.

Two Dimensional Trigonometric Problems

Concept Explanation: Involves solving problems on a flat surface (plane) using angles and distances. Often uses right-angled triangles and trig ratios.

Applications:

• Finding heights and distances (building, flagpole)
• Solving problems with bearings and directions.

Example: A ladder 5 m long rests against a wall making a 60° angle with the ground. Find the height (h) where the ladder touches the wall. Step 1: Draw the triangle and identify the required parts. Step 2: Use sine ratio.

\sin 60^\circ = \frac{h}{5}

h = 5 \sin 60^\circ

h = 5 \times 0,8660 = 4,33\, \text{m}

Common Mistake: Not drawing or labelling a clear triangle.

Exam Tip: Always draw diagrams for worded trigonometry questions.

Three Dimensional Trigonometric Problems

Concept Explanation: Applies trigonometry to 3D objects like pyramids, poles, and buildings. Involves working with triangles in three dimensions.

Applications:

• Calculating lengths, angles, or heights in pyramids, prisms, and rectangular boxes.

Example: Given a pyramid with a square base, calculate the slant height using the vertical height and half of the base.

Step 1: Use Pythagoras’ theorem in the triangular face. Let the vertical height h = 6 m, half the diagonal of base a = 4 m.

\text{Slant height} = \sqrt{6^2 + 4^2}

= \sqrt{36 + 16}

= \sqrt{52}

= 7,21\, \text{m}

Common Mistake: Confusing the base length with the diagonal or not applying 2D trig first to find intermediate lengths.

Exam Tip: Break the 3D problem into 2D triangles.

Sine Rule

Concept Explanation: The Sine Rule is used for non-right-angled triangles, relating the sides and their opposite angles.

Formula:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Example: In triangle PQR, PQ = 10 cm, \angle R = 45^\circ, QR = 8 cm, \angle P = 30^\circ. Find \angle Q. Step 1: Write the Sine Rule.

\frac{PQ}{\sin Q} = \frac{QR}{\sin P}

\frac{10}{\sin Q} = \frac{8}{\sin 30^\circ}

\frac{10}{\sin Q} = \frac{8}{0,5}

\frac{10}{\sin Q} = 16

\sin Q = \frac{10}{16}

Q = \sin^{-1}(0,625) = 38,68^\circ

Common Mistake: Not matching sides and their opposite angles.

Exam Tip: Use the Sine Rule when you have side-angle-side (not enclosed) or angle-side-angle conditions.

Cosine Rule

Concept Explanation: The Cosine Rule is applicable to non-right-angled triangles when you know either three sides or two sides and the included angle.

Formula:

a^2 = b^2 + c^2 - 2bc \cos A

Example: Given triangle ABC with AB = 8 cm, AC = 5 cm, \angle BAC = 60^\circ. Find BC. Step 1: Write the Cosine Rule.

BC^2 = AB^2 + AC^2 - 2(AB)(AC)\cos BAC

Substitute:

= 8^2 + 5^2 - 2 \times 8 \times 5 \times \cos 60^\circ

= 64 + 25 - 80 \times 0,5

= 89 - 40 = 49

BC = 7\, \text{cm}

Common Mistake: Using the wrong angle or not squaring the side.

Exam Tip: Use the Cosine Rule for side-angle-side (enclosed angle) or side-side-side.

Area Rule

Concept Explanation: The Area Rule allows you to find the area of any triangle using two sides and the included angle.

Formula:

\text{Area} = \frac{1}{2} ab \sin C

Example: Given triangle with a = 6 cm, b = 9 cm, C = 30^\circ. Find the area.

Step 1: Write the formula.

\text{Area} = \frac{1}{2} ab \sin C

= \frac{1}{2} \times 6 \times 9 \times \sin 30^\circ

= 27 \times 0,5

= 13,5\, \text{cm}^2

Common Mistake: Using the wrong angle, not the included angle.

Exam Tip: Only use the Area Rule when dealing with two sides and the included angle.

These notes are designed to help you quickly review major trigonometry concepts for Grade 12, with examples and explanations following the South African CAPS curriculum and standard exam conventions.