Trigonometric Identities
Fundamental identities, Pythagorean identities, and proving identities
lessons (10)
Reciprocal Identities
Learn the reciprocal trigonometric functions (cosecant, secant, cotangent) and their relationships to sine, cosine, and tangent.
Pythagorean Identities
Learn the three fundamental Pythagorean identities and how to derive and apply them.
Even and Odd Trigonometric Identities
Learn how cosine and secant are even functions while sine, tangent, cosecant, and cotangent are odd functions.
Cofunction Identities
Learn how sine and cosine, tangent and cotangent, secant and cosecant are related through complementary angles.
Double Angle Identities
Learn the double angle formulas for sine, cosine, and tangent to simplify expressions and solve equations.
Half Angle Identities
Learn how to find exact values of sine, cosine, and tangent for half angles using the half-angle formulas.
Quotient Identities
Learn why tangent equals sine over cosine and cotangent equals cosine over sine, where they are defined, and how to use them to simplify expressions.
Simplifying Trigonometric Expressions
Learn a step-by-step toolbox for simplifying trigonometric expressions: rewrite in sine and cosine, use Pythagorean identities, factor, combine fractions and cancel.
Proving Trigonometric Identities
Learn how to prove a trig identity step by step: work on one side, convert to sine and cosine, combine fractions, use the conjugate, and disprove false claims with one angle.
Sum and Difference Identities
Learn the sum and difference formulas for sine, cosine and tangent, find exact values like sin 75° and combine angles from different quadrants.
Trigonometric identities are equations involving trig functions that are true for all valid inputs. These identities simplify expressions, solve equations, and prove other identities. Mastering fundamental identities unlocks advanced trigonometry.
What Students Will Learn
- Use Pythagorean identities
- Apply quotient and reciprocal identities
- Use sum and difference formulas
- Apply double and half-angle formulas
- Verify trigonometric identities
Frequently Asked Questions
What are the Pythagorean identities?
$\sin^2\theta + \cos^2\theta = 1$, $1 + \tan^2\theta = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$. They are derived from the Pythagorean theorem.
How do I prove an identity?
Work on one side only to transform it into the other side. Common strategies: convert to sine and cosine, factor, find common denominators, multiply by conjugates.
What are the sum and difference formulas?
$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$. $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$. Note the sign switches!