Trigonometric identities: Set 1

Learning Objectives Recall and apply the reciprocal and quotient identities. State and use the three Pythagorean identities in their various forms. Simplify complex trigonometric expressions using fundamental identities. Prove basic trigonometric identities by manipulating one side of the equation to match the other. Express one trigonometric function in terms of another. Factor trigonometric expressions and combine trigonometric fractions. Ever wondered how video game engines create realistic lighting and shadows? 🎮 It involves complex calculations built on the foundational 'rules' of trigonometry we're about to explore! Trigonometric identities are equations that are true for all values of the variables involved. They are the fundamental rules of trigonome...

TermDefinitionExample Trigonometric IdentityAn equation involving trigonometric functions that is true for all values of the variable for which both sides of the equation are defined.The equation `sin²(θ) + cos²(θ) = 1` is an identity because it is true for any angle θ. Reciprocal IdentitiesIdentities that define the relationship between a trigonometric function and its multiplicative inverse.`csc(θ) = 1/sin(θ)`. If `sin(θ) = 1/2`, then `csc(θ) = 2`. Quotient IdentitiesIdentities that express one trigonometric function as a ratio of two others.`tan(θ) = sin(θ)/cos(θ)`. This shows the relationship between tangent, sine, and cosine. Pythagorean IdentitiesA set of three fundamental identities derived from applying the Pythagorean theorem to a right triangle in the unit circle.`1 + tan²(θ) =...

Reciprocal & Quotient Identities `csc(θ) = \frac{1}{sin(θ)}` | `sec(θ) = \frac{1}{cos(θ)}` | `cot(θ) = \frac{1}{tan(θ)}` `tan(θ) = \frac{sin(θ)}{cos(θ)}` | `cot(θ) = \frac{cos(θ)}{sin(θ)}` Use these identities as your first step to simplify expressions, typically by converting all functions into terms of sine and cosine. The Pythagorean Identities 1. `sin²(θ) + cos²(θ) = 1` 2. `1 + tan²(θ) = sec²(θ)` 3. `1 + cot²(θ) = csc²(θ)` These are crucial for substitutions, especially when you see squared trigonometric functions. Remember they can be rearranged, for example: `sin²(θ) = 1 - cos²(θ)`.