Trigonometric Identities & Complementary Angles Explained (JAMB/WAEC Maths, Trigonometry II)

Trigonometric identities are equations involving trigonometric ratios that hold true for all valid angle values, while complementary angles are pairs of angles that add up to 90°, with a special relationship between their trigonometric ratios.

This is Part 2 of our Trigonometry series, building on the basic ratios (SOHCAHTOA) from Part 1. This topic is a regular feature in JAMB and WAEC Mathematics, usually tested through simplification and proof-style questions. This lesson covers the key identities and complementary angle relationships with worked examples.

Quick takeaways

• A trigonometric identity is an equation involving trigonometric ratios that holds true for every valid value of the angle involved.
• The most fundamental identity is: sin²θ + cos²θ = 1, derived directly from Pythagoras’ theorem.
• Tangent can always be expressed in terms of sine and cosine: tan θ = sin θ / cos θ.
• Complementary angles are two angles that add up to 90°, and sin θ = cos(90° − θ) for any angle θ.
• Exam questions often ask you to simplify an expression using an identity, or to find a trigonometric ratio using the complementary angle relationship without a calculator.

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric ratios (sine, cosine, tangent) that remains true for every valid value of the angle involved, not just specific cases. Identities are used to simplify complex trigonometric expressions, prove other mathematical statements, or rewrite an expression in a more useful form for solving a problem.

What is the most fundamental trigonometric identity?

The most important and frequently used identity is the Pythagorean identity:

sin²θ + cos²θ = 1

This identity comes directly from applying Pythagoras’ theorem to a right-angled triangle with a hypotenuse of length 1, and it holds true for every angle θ. From this single identity, other related identities can also be derived, such as sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ, which are useful for rearranging expressions during simplification.

How is tangent related to sine and cosine?

Tangent can always be expressed as the ratio of sine to cosine:

tan θ = sin θ / cos θ

This relationship is useful for simplifying expressions that mix all three ratios, since it allows you to rewrite tangent entirely in terms of sine and cosine, reducing the number of different ratios you need to work with in a single problem.

What are complementary angles, and how do they relate to trigonometric ratios?

Complementary angles are two angles whose measures add up to 90°. In trigonometry, complementary angles have a special relationship: the sine of an angle equals the cosine of its complement, and vice versa:

sin θ = cos(90° − θ) and cos θ = sin(90° − θ)

For example, sin 30° = cos 60°, since 30° and 60° are complementary angles (30° + 60° = 90°). This relationship is frequently used to find a trigonometric ratio without a calculator, especially for well-known angle values like 30°, 45°, and 60°.

How do you use identities and complementary angles to simplify expressions?

A typical simplification question asks you to rewrite a more complex trigonometric expression using one or more identities. The general approach:

1. Look for sin²θ + cos²θ patterns, which can often be replaced with 1
2. Rewrite tangent in terms of sine and cosine if the expression mixes multiple ratios
3. Use complementary angle relationships if the expression involves an angle and its complement (i.e., involves both θ and 90° − θ)
4. Simplify algebraically once everything is expressed in consistent terms

Common mistakes students make with this topic

• Forgetting the Pythagorean identity and attempting to simplify sin²θ + cos²θ expressions from scratch each time
• Mixing up the complementary angle relationship, such as writing sin θ = sin(90° − θ) instead of sin θ = cos(90° − θ)
• Treating an identity as only true for one specific angle, rather than understanding it holds for all valid angles
• Forgetting that tan θ = sin θ / cos θ, and instead treating tangent as a completely separate, unrelated ratio

Frequently asked questions

Q: What is the Pythagorean identity in trigonometry? The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ, derived directly from applying Pythagoras’ theorem to a right-angled triangle with a hypotenuse of length 1.

Q: How is tangent related to sine and cosine? Tangent can always be expressed as the ratio of sine to cosine: tan θ = sin θ / cos θ.

Q: What are complementary angles? Complementary angles are two angles whose measures add up to 90°, and they have the special relationship that sin θ = cos(90° − θ).

Q: How can complementary angles help you find trigonometric ratios without a calculator? Since sin θ = cos(90° − θ), you can find the sine or cosine of certain angles by relating them to their complement, which is especially useful for well-known angle values like 30°, 45°, and 60°.

New to trigonometry? Start with Trigonometry Basics (SOHCAHTOA and Pythagoras) →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →