Trigonometric identities are equations involving sine, cosine, tangent, and related functions that are true for every angle where both sides are defined. They let you rewrite expressions into forms that are easier to calculate, graph, or solve. Identities matter in physics, engineering, and calculus because waves, rotations, oscillations, and circular motion are described with trigonometric functions.
The unit circle is the foundation: for an angle θ, the point on the circle is (cos θ, sin θ), so the x-coordinate gives cosine and the y-coordinate gives sine. A right-triangle overlay connects these same ideas to opposite, adjacent, and hypotenuse ratios. More advanced identities, such as sum and difference, double-angle, and half-angle formulas, come from combining angles and using the Pythagorean relationship sin^2 θ + cos^2 θ = 1.
Key Facts
- Unit circle coordinates: (x, y) = (cos θ, sin θ)
- Pythagorean identity: sin^2 θ + cos^2 θ = 1
- Quotient identity: tan θ = sin θ / cos θ, where cos θ ≠ 0
- Sum and difference: sin(a ± b) = sin a cos b ± cos a sin b
- Cosine sum and difference: cos(a ± b) = cos a cos b ∓ sin a sin b
- Double-angle identities: sin 2θ = 2 sin θ cos θ and cos 2θ = cos^2 θ - sin^2 θ
Vocabulary
- Identity
- An identity is an equation that is true for all allowed values of the variable.
- Unit circle
- The unit circle is a circle of radius 1 centered at the origin, used to define trig functions for any angle.
- Reference angle
- A reference angle is the acute angle between the terminal side of an angle and the x-axis.
- Quadrant signs
- Quadrant signs describe which trig functions are positive or negative in each quadrant of the coordinate plane.
- Half-angle formula
- A half-angle formula rewrites a trig function of θ/2 using a trig function of θ.
Common Mistakes to Avoid
- Using sin(a + b) = sin a + sin b, which is wrong because sine does not distribute over addition. Use sin(a + b) = sin a cos b + cos a sin b instead.
- Forgetting quadrant signs, which leads to the wrong sign when finding values from a reference angle. Always identify the quadrant before choosing the positive or negative value.
- Canceling terms incorrectly in fractions such as sin θ / (sin θ + cos θ), which is wrong because a term cannot be canceled from only part of a sum. Factor first or use identities before simplifying.
- Dropping domain restrictions, which can create false identities after dividing by expressions like cos θ or sin θ. State that the identity only applies where the denominator is not zero.
Practice Questions
- 1 Use identities to simplify: (1 - cos^2 θ) / sin θ. Assume sin θ ≠ 0.
- 2 Find the exact value of sin 75° using the sum formula with 75° = 45° + 30°.
- 3 If sin θ = 3/5 and θ is in Quadrant II, find cos θ and tan θ.
- 4 Explain why cos 2θ can be written as cos^2 θ - sin^2 θ, 1 - 2 sin^2 θ, or 2 cos^2 θ - 1, and describe when one form might be more useful than another.