5.6.1 Compound Angle Formulae

Compound angle formulae are trigonometric identities that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine, cosine, and tangent of the individual angles. These formulae are extremely useful in solving trigonometric equations, simplifying expressions, and proving other trigonometric identities.

1. Sine of a Sum/Difference:

Sine of the Sum of Two Angles:

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

This formula expresses the sine of the sum of two angles $A$ and $B$ as a combination of the sines and cosines of the individual angles.
Sine of the Difference of Two Angles:

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

This formula expresses the sine of the difference of two angles A and B similarly.

2. Cosine of a Sum/Difference:

Cosine of the Sum of Two Angles:

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

This formula expresses the cosine of the sum of two angles $A$ and $B$ in terms of the cosines and sines of the individual angles.
Cosine of the Difference of Two Angles:

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

This formula expresses the cosine of the difference of two angles $A$ and $B.$

3. Tangent of a Sum/Difference:

Tangent of the Sum of Two Angles:

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

This formula expresses the tangent of the sum of two angles $A$ and $B$ in terms of the tangents of the individual angles.
Tangent of the Difference of Two Angles:

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

This formula expresses the tangent of the difference of two angles $A$ and $B.$

4. Derivation of Compound Angle Formulae:

The compound angle formulae can be derived using the unit circle, trigonometric identities, and geometric methods. Here's a brief outline of how you might derive them:

Sine and Cosine Formulae:
Consider the angles A and B placed on the unit circle. Use the coordinates of the points where the terminal sides of these angles intersect the circle to express $sin(A + B)$ and $\cos(A + B).$
Use the fact that the length of the hypotenuse in the unit circle is $1$ , and apply the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to derive the formulae.
Tangent Formula:
The tangent formulae can be derived by dividing the sine formula by the cosine formula:

$\tan(A \pm B) = \frac{\sin(A \pm B)}{\cos(A \pm B)}$

5. Example Problems Using Compound Angle Formulae:

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Example 1: Calculate $\sin(75^\circ$ ) using the compound angle formula.

Solution:
Recognize that $75^\circ = 45^\circ + 30^\circ.$
Apply the sine of a sum formula:

$\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

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Example 2: Simplify $\cos(45^\circ - 30^\circ).$

Solution:
Recognize that $45^\circ - 30^\circ = 15^\circ.$
Apply the cosine of a difference formula:

$\cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ$

$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

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Example 3: Find $\tan(105^\circ)$ .

Solution:
Recognize that $105^\circ = 60^\circ + 45^\circ.$
Apply the tangent of a sum formula:

$\tan(105^\circ) = \frac{\tan 60^\circ + \tan 45^\circ}{1 - \tan 60^\circ \tan 45^\circ} = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1}$

$= \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}} = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})}$

$= \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{1 - 3} = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{-2} = \frac{-4\sqrt{3} - 2}{2} = \frac{-4 - 2\sqrt{3}}{-2}$

$= \frac{2 + \sqrt{3}}{2}$

6. Applications of Compound Angle Formulae:

Solving Trigonometric Equations: Compound angle formulae are essential when solving trigonometric equations involving sums or differences of angles.
Proving Identities: These formulae are frequently used in proving more complex trigonometric identities.
Simplifying Expressions: They help in simplifying trigonometric expressions to make them easier to evaluate or integrate.
Geometry and Physics: These formulae are useful in applications involving rotations, wave interference, and oscillations.

Summary:

infoNote

Compound angle formulae are powerful tools in trigonometry that allow you to express the sine, cosine, and tangent of sums and differences of angles in terms of the individual angles.
These formulae are essential for solving complex trigonometric equations, simplifying expressions, and proving identities.
Mastery of these identities expands your ability to tackle a wide range of trigonometric problems in mathematics, physics, and engineering.

Expanding Trig Brackets: Compound Angle Formulae

When expanding trigonometric functions, e.g., $\sin(A \pm B)$ , the ordinary rules of algebra do not apply.

Proposition:

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

Proof:

Finding the lengths of the four coloured sides:

For the side corresponding to $\cos A$ :

\cos A = \frac{\text{Red segment}}{1} \Rightarrow \text{Red segment} = \cos A

For the side corresponding to $\sin A$ :

\sin A = \frac{\text{Blue segment}}{1} \Rightarrow \text{Blue segment} = \sin A

For the side corresponding to $\sin B \cos A$ :

\cos B = \frac{\text{Green segment}}{\sin A} \Rightarrow \text{Green segment} = \sin A \cos B

For the side corresponding to $\sin B$ :

\sin B = \frac{\text{Orange segment}}{\sin A} \Rightarrow \text{Orange segment} = \sin B \cos A

For the equation $\sin(A + B)$ :

The calculation in the diagram confirms that:

\sin(A + B) = \frac{\text{purple segment}}{1} \Rightarrow \text{purple segment} = \sin(A + B)

From the diagram, we can also see that:

\therefore \quad \sin(A + B) \equiv \sin(A)\cos(B) + \sin(B)\cos(A) \quad \text{(:success[proven by the diagram])}

Proposition:

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Proof:

Diagram Explanation:

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The base of the blue triangle is given by:

\cos(A+B) = \frac{x}{1} \implies x = \cos(A+B)

Notice also that $x$ can be obtained by:

\cos A \cos B - \sin A \sin B

\therefore \cos(A+B) = \cos A \cos B - \sin A \sin B \quad \square

Compound Angle Formulae

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

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

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Example:

Use the compound angle formulae to expand and simplify $\sin(\theta + \frac{\pi}{2})$ :

Let $A = \theta, B = \frac{\pi}{2}$ :

\sin(\theta + \frac{\pi}{2}) = \xcancel{\sin \theta \cos \frac{\pi}{2}} + \cos \theta \sin \frac{\pi}{2}

Since $\cos \frac{\pi}{2} = 0$ and $\sin \frac{\pi}{2} = 1$ :

= \cos \theta

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Example: "Factorise"

\sin 45 \cos 15 - \cos 45 \sin 15

Hint: The trick here is to look at which trig identity it most closely matches.

\sin(A - B) \equiv \sin A \cos B - \cos B \sin A

We can see here that $A = 45^\circ, B = 15^\circ$ .

\Rightarrow \text{it is identical to } \sin(45^\circ - 15^\circ) = \sin 30^\circ

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Given that $\sin A = \frac{4}{5}, 0^\circ < A < 90^\circ$ and that $\cos B = \frac{2}{3}, 0^\circ < B < 90^\circ$ , find without using a calculator the value of:

$\tan A$
$\sin B$
$\cos (A + B)$
$\sin (A + B)$

Diagrams:

For angle A: $\sin A=\frac {4}{5}=\frac {OPP}{HYP}$

Hypotenuse = 5, Opposite = 4, Adjacent = $\sqrt{5^2 - 4^2} = \highlight[3]$

For angle B: $\cos B=\frac {2}{3}=\frac {ADJ}{HYP}$

Hypotenuse = 3, Adjacent = 2, Opposite = $\sqrt{3^2 - 2^2} = \highlight[\sqrt{5}]$

Solutions:

$a) \tan A = \frac{\text{OPP}}{\text{ADJ}} = \frac{4}{3}$

b) $\sin B = \frac{\text{OPP}}{\text{HYP}} = \frac{\sqrt{5}}{3}$

c) $\cos(A + B) = \cos A \cos B - \sin A \sin B$

We can now find each of these individually from the triangles to evaluate the expression. From above triangle, $\cos A = \frac{\text{ADJ}}{\text{HYP}} = \frac{3}{5}$

\therefore \cos(A + B) = \left(\frac{\cancel3}{5}\right)\left(\frac{2}{\cancel3}\right) - \left(\frac{4}{5}\right)\left(\frac{\sqrt{5}}{3}\right) = \frac{2}{5} - \frac{4\sqrt{5}}{15} = \frac{6 - 4\sqrt{5}}{15}

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

= \left(\frac{4}{5}\right)\left(\frac{2}{3}\right) + \left(\frac{\cancel3}{5}\right)\left(\frac{\sqrt{5}}{\cancel3}\right) = \frac{8}{15} + \frac{\sqrt{5}}{5} = \frac{8 + 3\sqrt{5}}{15}

Solving Equations With Compound Angle Formulae

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Example: Solve $\cos \theta \cos(0.2^\circ) - \sin \theta \sin(0.2^\circ) = 0.4$ For $0 \leq \theta \leq 2\pi$

In any example like this, there are certain questions to ask:

Is it solvable as it is? If so, do it.
If not, ask "do I need to expand or factorize?"

Step-by-Step Solution:

Let $A = \theta$ and $B = 0.2^\circ$

\Rightarrow \cos(A + B) = \cos(\theta + 0.2^\circ)

\Rightarrow \cos(\theta + 0.2^\circ) = 0.4

From the cosine graph:

\theta + 0.2 = 1.1593

2\pi - 1.1593

2\pi + 1.1593

0.2 \leq \theta + 0.2 \leq 2\pi + 0.2

\theta +0.2 = 1.1593, 5.1239, \cancel{7.4425}

Calculating for $\theta$ :

\theta = :success[0.9533], :success[4.9239]

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Example: Solve $\cos(x + 30^\circ) = \sin x \quad for \quad 0 \leq x \leq 360^\circ$ .

\cos x \cos 30^\circ - \sin x \sin 30^\circ = \sin x

\frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x = \sin x

\frac{\sqrt{3}}{2} \cos x = \frac{3}{2} \sin x

\cos x = \sqrt{3} \sin x

1 = \sqrt{3} \frac{\sin x}{\cos x}

1 = \sqrt{3} \tan x \Rightarrow \tan x = \frac{1}{\sqrt{3}}

x = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = 30^\circ, 210^\circ

From the graph:

x = :success[30^\circ], :success[210^\circ]

Compound Angle Formulae Simplified Revision Notes for A-Level OCR Maths Pure

5.6.1 Compound Angle Formulae

1. Sine of a Sum/Difference:

2. Cosine of a Sum/Difference:

3. Tangent of a Sum/Difference:

4. Derivation of Compound Angle Formulae:

5. Example Problems Using Compound Angle Formulae:

Example 1: Calculate $\sin(75^\circ$ ) using the compound angle formula.

Example 2: Simplify $\cos(45^\circ - 30^\circ).$

Example 3: Find $\tan(105^\circ)$ .

6. Applications of Compound Angle Formulae:

Summary:

Expanding Trig Brackets: Compound Angle Formulae

Finding the lengths of the four coloured sides:

Proposition:

Diagram Explanation:

Compound Angle Formulae

Example:

Example: "Factorise"

Given that $\sin A = \frac{4}{5}, 0^\circ < A < 90^\circ$ and that $\cos B = \frac{2}{3}, 0^\circ < B < 90^\circ$ , find without using a calculator the value of:

Diagrams:

Solutions:

Solving Equations With Compound Angle Formulae

500K+ Students Use These Powerful Tools to Master Compound Angle Formulae For their A-Level Exams.

Other Revision Notes related to Compound Angle Formulae you should explore

Compound Angle Formulae Simplified Revision Notes for A-Level OCR Maths Pure

5.6.1 Compound Angle Formulae

1. Sine of a Sum/Difference:

2. Cosine of a Sum/Difference:

3. Tangent of a Sum/Difference:

4. Derivation of Compound Angle Formulae:

5. Example Problems Using Compound Angle Formulae:

Example 1: Calculate sin⁡(75∘\sin(75^\circsin(75∘) using the compound angle formula.

Example 2: Simplify cos⁡(45∘−30∘).\cos(45^\circ - 30^\circ).cos(45∘−30∘).

Example 3: Find tan⁡(105∘)\tan(105^\circ)tan(105∘).

6. Applications of Compound Angle Formulae:

Summary:

Expanding Trig Brackets: Compound Angle Formulae

Finding the lengths of the four coloured sides:

Proposition:

Diagram Explanation:

Compound Angle Formulae

Example:

Example: "Factorise"

Given that sin⁡A=45,0∘<A<90∘\sin A = \frac{4}{5}, 0^\circ < A < 90^\circsinA=54​,0∘<A<90∘ and that cos⁡B=23,0∘<B<90∘\cos B = \frac{2}{3}, 0^\circ < B < 90^\circcosB=32​,0∘<B<90∘, find without using a calculator the value of:

Diagrams:

Solutions:

Solving Equations With Compound Angle Formulae

500K+ Students Use These Powerful Tools to Master Compound Angle Formulae For their A-Level Exams.

Other Revision Notes related to Compound Angle Formulae you should explore

Example 1: Calculate $\sin(75^\circ$ ) using the compound angle formula.

Example 2: Simplify $\cos(45^\circ - 30^\circ).$

Example 3: Find $\tan(105^\circ)$ .

Given that $\sin A = \frac{4}{5}, 0^\circ < A < 90^\circ$ and that $\cos B = \frac{2}{3}, 0^\circ < B < 90^\circ$ , find without using a calculator the value of: