Warm Up Find the reciprocal of each integer: 6 7 1 2 7 1 8 5 4

Using Definitions of the Trigonometric Functions Section 1.4 Using Definitions of the Trigonometric Functions SWBAT: Give the signs of the six trigonometric functions for a given angle. Identify the quadrant or quadrants for angles satisfying given conditions.

Reciprocal Identities

Using the Reciprocal Identities Find each function value. Find cos  if sec  = Find sin  if csc  Find cos  if sec  = 5 3 Find sin θ if csc θ = −√12 2

Signs & Ranges of Function Values Remember: r is the distance from the origin to a point (x, y) . Distance is never negative so r >0. If we find the six trigonometric functions of an angle θ in quadrant I, (x, y) are both positive an so are all 6 fucntions. What happens if we have a point in a different quadrant?!?

Signs & Ranges of Function Values All Students Take Calculus

Signs & Ranges of Function Values  in Quadrant sin  cos  tan  cot  sec  csc  I + + + + + + +  II  + III  + IV

Identifying Quadrants Identify the quadrant (or quadrants) of any angle θ that satisfies: sin θ > 0 and tan θ < 0 cos  > 0 and sec  < 0 sin  > 0 and cos  < 0 csc  < 0 and sec  < 0 tan > 0

Identifying Quadrants Give the signs of the six trigonometric functions for each of the following angles: 74  183  302  406  -121 

Ranges of Trigonometric Functions

DECIDING WHETHER A VALUE IS IN THE RANGE OF A TRIGONOMETRIC FUNCTION Decide whether each statement is possible or impossible. (a) sin θ = 2.5 Impossible (b) tan θ = 110.47 Possible (c) sec θ = .6 Impossible

FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Suppose that angle  is in quadrant II and Find the values of the other five trigonometric functions. Let r = 3. Then y = 2. What is x? Use the Pythagorean Theorem… Since  is in quadrant II,

Let r =3, y = 2, x = √5 Sin θ = Csc θ = Cos θ = Sec θ = FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Let r =3, y = 2, x = √5 Sin θ = Csc θ = Cos θ = Sec θ = Tan θ = Cot θ = Remember to rationalize the denominator.

FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Suppose that angle  is in quadrant II and Find the values of the other five trigonometric functions. Cos θ= −√3 4 Cos θ= −√3 4 Let r = 4. Then x = -√3 What is y? Use the Pythagorean Theorem…

FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Sin θ = Csc θ = Cos θ = Sec θ = Tan θ = Cot θ =

Homework Pages 34-35 # 4, 6, 28, 32, 34, 40, 48, 50