Compound Angles

Overview

Compound angles are angles made by adding or subtracting simpler angles. Instead of treating $75^\circ$ as a mysterious calculator value, a learner can see it as $45^\circ + 30^\circ$. Instead of treating $15^\circ$ as new, a learner can see it as $45^\circ - 30^\circ$.

This chapter builds formulas for expressions such as $\sin(A+B)$, $\cos(A-B)$, and $\tan(A+B)$. The goal is not to memorize symbols blindly. The goal is to know which identity fits the expression, substitute carefully, and simplify without losing signs.

Compound-angle work connects earlier exact values from Trigonometric Ratios with later algebraic manipulation and exam-style trigonometry. It also supports identities, proof questions, and exact-value questions that should not need a trigonometric table.

Prerequisites

• Trigonometric Ratios - Learners need exact values for $30^\circ$, $45^\circ$, and $60^\circ$.
• Basic algebra - Identities require substitution, expansion, factorization, and simplification.
• Fractions and radicals - Exact answers often include $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$, and products of surds.
• Angle facts - Learners should recognize complementary, supplementary, and special angles.
• Calculator discipline - Calculators can check values, but exact-value work should be done symbolically.
• Sine Rule and Cosine Rule - Some advanced triangle work uses exact trigonometric values that compound angles help produce.

Learning Scope

This chapter covers addition and subtraction identities for sine, cosine, and tangent; exact values of angles such as $15^\circ$, $75^\circ$, and $105^\circ$; double-angle identities as a natural special case; identity proof routines; and common sign checks.

This page does not attempt a full study of trigonometric graphs, radian measure, or advanced calculus identities. It focuses on Form IV learner use: expand correctly, simplify exactly, and explain each transformation.

Subtopics

What A Compound Angle Means

A compound angle is formed by combining two angles:

$$ A + B $$

or:

$$ A - B $$

For example:

$$ 75^\circ = 45^\circ + 30^\circ $$

and:

$$ 15^\circ = 45^\circ - 30^\circ $$

This matters because $30^\circ$, $45^\circ$, and $60^\circ$ have exact trigonometric values. By combining them, we can find exact values for other angles.

Sine Of A Sum

The identity for sine of a sum is:

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

The sign in the middle stays positive when the angle is a sum.

Key insight: Sine of a sum is not $\sin A + \sin B$. The identity uses products of sine and cosine.

Sine Of A Difference

The identity for sine of a difference is:

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

The sign in the middle follows the sign in the angle expression.

For example:

$$ \sin(45^\circ - 30^\circ) = \sin45^\circ\cos30^\circ - \cos45^\circ\sin30^\circ $$

Cosine Of A Sum

The identity for cosine of a sum is:

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

Warning sign: Cosine of a sum has a minus sign in the middle. This is a common place for errors because sine of a sum uses plus.

Cosine Of A Difference

The identity for cosine of a difference is:

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

For cosine, the middle sign changes from the sign in the bracket:

• $\cos(A+B)$ uses minus.
• $\cos(A-B)$ uses plus.

Key insight: Cosine is the identity where the sign feels opposite to many learners. Slow substitution prevents mistakes.

Tangent Of A Sum And Difference

The tangent identities are:

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

and:

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

These formulas are useful when the problem gives tangent values directly or asks for exact tangent values.

Warning sign: The denominator sign changes in the opposite direction from the numerator sign.

Double-Angle Identities

Double-angle identities come from setting $B = A$.

For sine:

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

For cosine:

$$ \cos 2A = \cos^2 A - \sin^2 A $$

Using $\sin^2 A + \cos^2 A = 1$, cosine can also be written as:

$$ \cos 2A = 2\cos^2 A - 1 $$

or:

$$ \cos 2A = 1 - 2\sin^2 A $$

For tangent:

$$ \tan 2A = \frac{2\tan A}{1-\tan^2 A} $$

Exact Values From Compound Angles

Many exact values can be found by choosing special angles.

For example:

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

This uses:

$$ \sin45^\circ = \frac{\sqrt{2}}{2}, \quad \cos45^\circ = \frac{\sqrt{2}}{2} $$

and:

$$ \sin30^\circ = \frac{1}{2}, \quad \cos30^\circ = \frac{\sqrt{3}}{2} $$

Exact-value work rewards careful fraction handling more than speed.

Proving Identities

To prove a compound-angle identity:

1. Start from the more complicated side.
2. Replace compound expressions using known identities.
3. Use basic identities such as $\sin^2 A + \cos^2 A = 1$.
4. Simplify one line at a time.
5. Stop when the target expression is reached.

Key insight: Do not change both sides at once unless each side is clearly labelled. In exam solutions, working from one side is usually easier to follow.

Checking Signs And Quadrants

Compound angles can produce angles greater than $90^\circ$, such as:

$$ 105^\circ = 60^\circ + 45^\circ $$

Since $105^\circ$ is in the second quadrant:

• $\sin 105^\circ$ should be positive.
• $\cos 105^\circ$ should be negative.
• $\tan 105^\circ$ should be negative.

Even when a learner does not formally study all quadrant rules in depth, this sign check helps catch errors.

Key Terms

• Compound angle: An angle written as the sum or difference of two angles.
• Identity: An equation true for all allowed values of the variable.
• Exact value: A trigonometric value written using fractions and radicals instead of decimals.
• Sum identity: A formula for a trigonometric function of $A+B$.
• Difference identity: A formula for a trigonometric function of $A-B$.
• Double-angle identity: A formula for a trigonometric function of $2A$.
• Surd: A radical expression such as $\sqrt{2}$ or $\sqrt{3}$.
• Simplification: Rewriting an expression in a cleaner equivalent form.
• Quadrant check: A sign check based on where an angle lies on the coordinate plane.
• Undefined tangent: A tangent value whose denominator becomes zero.

Worked Examples

Example 1: Find $\sin 75^\circ$ Exactly

Step 1: Write $75^\circ$ as a sum of special angles.

$$ 75^\circ = 45^\circ + 30^\circ $$

Step 2: Use the sine sum identity.

$$ \sin(45^\circ + 30^\circ) = \sin45^\circ\cos30^\circ + \cos45^\circ\sin30^\circ $$

Step 3: Substitute exact values.

$$ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$

Step 4: Simplify.

$$ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} $$

$$ \sin75^\circ = \frac{\sqrt{6}+\sqrt{2}}{4} $$

Check: $75^\circ$ is close to $90^\circ$, so its sine should be close to $1$. The value is about $0.966$, which is reasonable.

Example 2: Find $\cos 75^\circ$ Exactly

Step 1: Use the cosine sum identity.

$$ \cos75^\circ = \cos(45^\circ + 30^\circ) $$

$$ = \cos45^\circ\cos30^\circ - \sin45^\circ\sin30^\circ $$

Step 2: Substitute exact values.

$$ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)

• \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)

$$

Step 3: Simplify.

$$ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} $$

$$ \cos75^\circ = \frac{\sqrt{6}-\sqrt{2}}{4} $$

Check: $75^\circ$ is close to $90^\circ$, so its cosine should be small and positive. The expression gives about $0.259$, which fits.

Example 3: Find $\tan 15^\circ$ Exactly

Step 1: Write $15^\circ$ as a difference.

$$ 15^\circ = 45^\circ - 30^\circ $$

Step 2: Use the tangent difference identity.

$$ \tan(45^\circ - 30^\circ) = \frac{\tan45^\circ - \tan30^\circ}{1+\tan45^\circ\tan30^\circ} $$

Step 3: Substitute exact values.

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

Step 4: Clear the small fractions by multiplying numerator and denominator by $\sqrt{3}$.

$$ = \frac{\sqrt{3}-1}{\sqrt{3}+1} $$

Step 5: Rationalize if required.

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

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

$$ \tan15^\circ = 2 - \sqrt{3} $$

Check: $15^\circ$ is small, so tangent should be positive and less than $1$. Since $2-\sqrt{3} \approx 0.268$, the answer is sensible.

Example 4: Use A Double-Angle Identity

If $\sin A = \frac{3}{5}$ and $A$ is acute, find $\sin 2A$.

Step 1: Find $\cos A$ using a right-triangle or identity check.

Since $A$ is acute:

$$ \cos A = \frac{4}{5} $$

Step 2: Use the sine double-angle identity.

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

Step 3: Substitute.

$$ \sin 2A = 2\left(\frac{3}{5}\right)\left(\frac{4}{5}\right) $$

$$ \sin 2A = \frac{24}{25} $$

Check: The answer is less than $1$, so it is possible as a sine value.

Example 5: Prove A Simple Identity

Prove that:

$$ \cos(A-B) - \cos(A+B) = 2\sin A\sin B $$

Step 1: Start with the left-hand side.

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

Step 2: Expand each compound angle.

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

• (\cos A\cos B - \sin A\sin B)

$$

Step 3: Remove brackets carefully.

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

Step 4: Cancel and collect like terms.

$$ = 2\sin A\sin B $$

Therefore:

$$ \cos(A-B) - \cos(A+B) = 2\sin A\sin B $$

Common Mistakes

• Writing $\sin(A+B)$ as $\sin A + \sin B$.
• Using the sine sign pattern for cosine identities.
• Forgetting that $\cos(A+B)$ has a minus sign.
• Forgetting that $\cos(A-B)$ has a plus sign.
• Reversing the denominator sign in tangent identities.
• Substituting decimal approximations when an exact answer is required.
• Dropping brackets when subtracting $\cos(A+B)$ or another expanded expression.
• Treating $\sin^2 A$ as $\sin(A^2)$.
• Forgetting to check whether a final tangent value should be positive or negative.
• Rationalizing incorrectly by multiplying only the denominator.

Practice Tasks

Foundation

1. Write $75^\circ$ as a sum of two special angles.
2. Write $15^\circ$ as a difference of two special angles.
3. State the identity for $\sin(A+B)$.
4. State the identity for $\cos(A-B)$.
5. Decide whether $\cos(A+B)$ uses a plus sign or minus sign between the two products.
6. Explain why $\sin(A+B)$ is not the same as $\sin A + \sin B$.

Skill-Building

1. Find $\sin15^\circ$ exactly.
2. Find $\cos15^\circ$ exactly.
3. Find $\sin105^\circ$ exactly.
4. Find $\cos105^\circ$ exactly and check that the sign is reasonable.
5. Find $\tan75^\circ$ exactly.
6. If $\sin A = \frac{5}{13}$ and $A$ is acute, find $\cos A$ and then $\sin 2A$.
7. If $\cos A = \frac{12}{13}$ and $A$ is acute, find $\cos 2A$.
8. Expand $\sin(x+30^\circ)$ using compound-angle identities.

Exam-Style

1. Without using a table, find the exact value of $\sin75^\circ$.
2. Without using a table, find the exact value of $\cos15^\circ$.
3. Prove that $\sin(A+B) + \sin(A-B) = 2\sin A\cos B$.
4. Prove that $\cos(A-B) + \cos(A+B) = 2\cos A\cos B$.
5. If $\tan A = \frac{3}{4}$ and $A$ is acute, find $\tan 2A$.
6. Show that $\cos^2 A - \sin^2 A = 1 - 2\sin^2 A$.
7. Given $\sin A = \frac{7}{25}$ for acute $A$, find $\cos 2A$.
8. Simplify $\cos(60^\circ+x) + \cos(60^\circ-x)$.

Challenge

1. Derive $\tan(A+B)$ from $\tan(A+B)=\frac{\sin(A+B)}{\cos(A+B)}$.
2. Prove that $\sin(A+B)\sin(A-B)=\sin^2 A-\sin^2 B$.
3. Find an exact expression for $\tan105^\circ$ and explain why its sign is negative.
4. Use compound-angle identities to show that $\sin75^\circ=\cos15^\circ$.
5. Create an identity proof where the first line requires expanding both $\cos(A-B)$ and $\cos(A+B)$, then solve it.
6. Find all values of $A$ in a stated interval for which the denominator of $\tan 2A$ is zero.

Generated Question Layer

Future generated practice questions for this topic should include:

• Formula-recall questions that distinguish sine, cosine, and tangent sign patterns.
• Exact-value questions for $15^\circ$, $75^\circ$, $105^\circ$, and related angles.
• Stepwise substitution tasks using $30^\circ$, $45^\circ$, and $60^\circ$ values.
• Identity-proof questions requiring expansion and simplification from one side.
• Error-analysis questions involving $\sin(A+B)=\sin A+\sin B$, sign slips, and missing brackets.
• Double-angle questions from given sine, cosine, or tangent values.
• Quadrant-check questions where learners predict the sign before simplifying.
• Calculator-check questions where decimal approximations are used only after exact work.

Generated questions should be original learner practice. They should not copy past-paper wording, even when they are calibrated to similar skills.

Learner Aid Opportunities

• chart: Display the six main sum and difference identities with sign warnings.
• diagram: Show how $75^\circ$ can be decomposed into $45^\circ+30^\circ$ on an angle ray.
• graph: Compare exact and decimal values for $\sin15^\circ$, $\sin75^\circ$, $\cos15^\circ$, and $\cos75^\circ$.
• animation: Expand $\sin(A+B)$ term by term, highlighting where each product comes from.
• interactive: Let learners choose an identity, substitute exact special-angle values, and simplify step by step.
• video: Work through an identity proof with brackets, cancellation, and final comparison.
• LLM tutor: Check a learner's expansion for sign errors and ask for a quadrant reasonableness check.

Exam-Derived Signals

The automatic 2021-2025 topic-frequency file lists topic-compound-angles in the Form IV trigonometry hub, but no reviewed past-question mapping is attached to this exact topic in the current extraction layer. Some trigonometry questions in the broader hub involve identities or exact values, but they are not reliable exact-topic links until reviewed.

The 2022 examination-format crosswalk is an official-format-derived planning source, but the crosswalk record itself is marked needs_manual_review. It groups this topic under Trigonometry and Pythagoras theorem with related trigonometry topics.

| Source | Signal | Status | Review caution | | --- | --- | --- | --- | | data/topic_frequency_2021_2025.json | Lists topic-compound-angles as a Form IV trigonometry topic. | Unreviewed extraction signal | Indicates topic presence, not a verified question trend. | | data/question_map_2021_2025.jsonl | No reliable exact-topic question records were found for topic-compound-angles in the inspected mapping. | Unreviewed absence signal | Broader trigonometry identity signals may exist but need manual paper review before being attached here. | | data/exam_format_topic_crosswalk_2022.jsonl | Format group Trigonometry and Pythagoras theorem maps to topic-trigonometric-[[ratio|ratios]], topic-angles-of-[[elevation|elevation]]-and-depression, topic-sine-rule-and-cosine-rule, and topic-compound-angles. | Official-format crosswalk, record marked needs_manual_review | The crosswalk includes unmatched term Pythagoras theorem, so topic grouping needs reviewer confirmation. |

Source And Review Notes

• Official syllabus status: The topic identity, Form IV placement, competence, and hub come from data/curriculum_map.json, which cites raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Official syllabus reference page: wiki/sources/csee-mathematics-syllabus-2023.md records the syllabus publisher, level, curriculum role, and assessment context.
• Assessment signal sources: data/topic_frequency_2021_2025.json, data/question_map_2021_2025.jsonl, and data/exam_format_topic_crosswalk_2022.jsonl.
• Review status: This is an unreviewed learner expansion from an official syllabus topic. The prose, examples, and practice tasks are original and should be checked by a mathematics reviewer before being marked reviewed.
• Exam-derived signals remain explicitly unreviewed unless a reviewer checks them against the original examination papers and marking schemes.
• Media status: This page uses text-only learner content. Future visuals should be created through reviewed learner-aid opportunities rather than embedded media files.
• Renderer QA: This page uses portable $...$ and $$...$$ math notation for future Obsidian, KaTeX, or MathJax rendering.