In this unit, you'll explore the power and beauty of trigonometric equations and identities, which allow you to express and relate different aspects of triangles, circles, and waves. You'll learn how to use trigonometric functions, their inverses, and various identities to solve and check equations and inequalities, and to model and analyze problems involving periodic motion, sound, light, and.

The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity. Using the Sum and Difference Formulas for Cosine

Formula Graph Derivative Table Example What is Sine Function? In a right-angle triangle, a sine function of an angle θ is equal to the opposite side to θ divided by hypotenuse. Sin θ = Opposite side/Hypotenuse This is the basic formula for sine function. See the figure below. Sin θ = Opposite / Hypotenuse What is Inverse Sine Function?

We may also derive the formula for the derivative of the inverse by first recalling that x = f (f − 1(x)). Then by differentiating both sides of this equation (using the chain rule on the right), we obtain 1 = f′ (f − 1(x)) (f − 1)′ (x)). Solving for (f − 1)′ (x), we obtain (f − 1)′ (x) = 1 f′ (f − 1(x)).

A right triangle with sides relative to an angle at the point. Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that.