1. 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. …
6.3924…

2. PYTHAGOREAN, RECIPROCAL, AND QUOTIENT PROPERTIES

3. 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.

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

5. 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.

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

7. 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.

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

9. 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

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

11. 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

12. 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.

13. 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.

14. 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.

15. 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:

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