True for every angle

The idea

An equation involving trigonometric ratios of an angle is a trigonometric identity if it is true for all values of the angle(s) involved.

Equation or identity?

In algebra, x + 2 = 5 is true only when x = 3. But (x + 1)² = x² + 2x + 1 is true for every value of x.

The second kind is called an identity. Trigonometry has equations of both kinds, and we need to tell them apart.

Which of these is true for every value of x?

• x + 2 = 5
• (x + 1)² = x² + 2x + 1
• x² = 9

Test the equations

Move the angle A and watch both sides of the equation. Try at least three different angles, then decide.

Identity: true for all angles

An equation involving trigonometric ratios of an angle is a trigonometric identity if it is true for all values of the angle(s) involved.

One angle where the two sides differ is enough to show that an equation is not an identity. But equal sides at a few angles do not prove an identity. A proof must work for every angle. In the next lessons we prove identities from the sides of a right triangle.

Notes

A trigonometric identity is an equation of trigonometric ratios that is true for all values of the angle involved. One mismatch shows it is not an identity.

Check yourself

Which of these is an identity?

For A = 30°, sin A = 1/2 and cos A = √3/2. What does this tell us about sin A = cos A?

Look at sin A + cos A = 1 at A = 45°. What is the left side, as a decimal to two places? (sin 45° = cos 45° = 1/√2 ≈ 0.707)

Answer: 1.41

sin 45° + cos 45° = 1/√2 + 1/√2 = √2 ≈ 1.41, which is not 1. So sin A + cos A = 1 is not an identity, although it is true at A = 0° and A = 90°.

An equation has equal sides at 30°, 45° and 60°. Does this prove it is an identity?

• x + 3 = 7. This is true only for x = 4.
• 2(x + 1) = 2x + 2 — correct. Yes! Both sides are the same for every value of x.
• x² = 9. This is true only for x = 3 (and x = −3).
• It is an identity.. The two sides are different at 30°, so it is not true for every angle.
• It is not an identity, because the sides differ at 30°. — correct. Yes! One angle with different sides is enough. (They are equal only at 45°.)
• Nothing: one angle cannot tell us anything.. One angle can show that an equation is false. 1/2 and √3/2 are not equal.
• Yes, three angles are enough.. Three angles do not cover every angle. Some other angle may give different sides.
• No. We must prove it for every angle, for example from the sides of a right triangle. — correct. Yes! Checking a few angles can only disprove an equation. A proof covers all angles.
• Yes, because 30°, 45° and 60° are special angles.. They are special for their exact values, but they are still only three of the angles.