WARM UP 1. What is the exact value of cos 30°? 2. What is the exact value of sin (π/4)? 3. What is the exact value of tan 60° 4.Write cos 57° in decimal form. 5. Write csc 9° in decimal form. 0.899866… 6.3924…

PYTHAGOREAN, RECIPROCAL, AND QUOTIENT PROPERTIES

OBJECTIVES Derive algebraically three kinds of properties expressing relationships among trigonometric functions Understand that the argument of x is used for both degrees and radians Understand how to graph the functions y = sec x, y = csc x and y = cot x based on graphs of y = cos x, y = sin x and y = tan x, respectively.

KEY TERMS & CONCEPTS Reciprocal properties Quotient properties Pythagorean properties Confunction Dual

INTRODUCTION In the exploration, you discovered the Pythagorean property: for all values of x. You also know that the secant, cosecant, and cotangent are reciprocals of cosine, sine, and tangent, respectively. In this section, you will prove these properties algebraically, along with the quotient properties, such as Because the properties you’ll learn in this section apply to all trigonometric functions, the argument x will be used for both degrees and radians.

RECIPROCAL PROPERTIES Recall that secant, cosecant and cotangent functions are the reciprocals of cosine, sine and tangent. because in the reference triangle and

RECIPROCAL PROPERTIES This relationship between secant and cosine is called a reciprocal property. The graphs show each y-value for the secant graphs is the reciprocal of the corresponding y-value for the cosine graph. Because , it follows that The asymptotes for the graph of the secant function occur at Where the value of the cosine function is zero.

PROPERTIES The Reciprocal Properties The domain excludes those values of x that produce a denominator equal to zero.

QUOTIENT PROPERTIES If you divide sin x by cos x, an interesting result appears Definition of sine and cosine Multiply the numerator by the reciprocal of the denominator. Definition of tangent Transitivity and symmetry

QUOTIENT PROPERTIES This relationship is called a quotient property. If you plot: The graphs of y1 and y2 will be superimposed.

QUOTIENT PROPERTIES Because cotangent is the reciprocal of tangent, another quotient property is Each of these quotient properties can be expressed in terms of secant and cosecant. For instance, Use the reciprocal properties for sine and cosine Simplify

THE QUOTIENT PROPERTIES Using the reciprocal property for cotangent gives THE QUOTIENT PROPERTIES Domain , where n is an integer. Domain , where n is an integer.

PYTHAGOREAN PROPERTIES The graph shows an arc of length x in standard position on the unit circle in a uv-coordinate system. By the Pythagorean theorem, point (u, v) at the endpoint of arc has the property: This property is true even if x terminates in a quadrant where u or v is negative because squares of negative numbers are the same as the squares of their absolute values.

PYTHAGOREAN PROPERTIES By definitions of cosine and sine, u = cos x and v = sin x Substitution into the equation gives the Pythagorean property for sine and cosine.

PYTHAGOREAN PROPERTIES The two other Pythagorean properties can be derived from this one. Start with the Pythagorean property for cosine and sine. Divide both sides of the equation by Dividing by instead of by results in the property:

PROPERTIES THE THREE PYTHAGOREAN PROPERTIES Domain: All real values of x. Domain , where n is an integer. Domain , where n is an integer.