1.4 Using the Definitions of the Trig Functions

Reciprocal Identities

Examples

Find cot θ, given tan θ = 5

Find sin θ, given csc θ =

Find cos θ, given sec θ = 9.80425133

Signs and Ranges of Trig Functions

The sign of a trig function depends on the quadrant (x,y) since r > 0

Memory aid - All Students Take Calculus

Determine the signs of the trig functions of an angle in standard position with the given measure:

54^o

260 ^o

- 60 ^o

Identifying the Quadrant of an Angle

Identify the quadrant (or possible quadrants) of θ which satisfies the given conditions

a) tan θ > 0, csc θ < 0

b) sin θ > 0, csc θ > 0

Ranges of Trig Functions

Trig Function	Range	Range in Interval Notation
sin θ, cos θ	-1 < y < 1	[-1, 1]
tan θ, cot θ	All real numbers	(-∞, ∞)
sec θ, csc θ	y < -1 or y > 1	(-∞,-1] Ù [1,∞)

Indicate possible or impossible:

a) cot θ = -.999 b) cos θ = -1.7 c) csc θ = 0

Finding All Function of θ Values given one value and the quadrant

Example

θ is in quadrant III and , find the other five function values

Pythagorean Identities

Pythagorean Identities

Quotient Identities

For all angles θ where the denominator is not zero

Use the quotient identities and Pythagorean Identities to find:

Find cos θ and tan θ, given

(find cos with Pythagorean)

(find tan with Quotient)

Note: Be careful with to choose the appropriate sign.

Find sin θ and cos θ, given that and θ is in Quadrant III