The Trigonometric Functions What about angles greater than 90°? 180°? The trigonometric functions are defined in terms of a point on a terminal side r is found by using the Pythagorean Theorem:

The 6 Trigonometric Functions of angle  are:

The Trigonometric Functions The trigonometric values do not depend on the selected point – the ratios will be the same:

First Quadrant: sin  = + cos  = + tan  = + csc  = + sec  = + cot  = +

Second Quadrant: sin  = + cos  = - tan  = - csc  = + sec  = - cot  = -

Third Quadrant: sin  = - cos  = - tan  = + csc  = - sec  = - cot  = + y x

Fourth Quadrant: sin  = - cos  = + tan  = - csc  = - sec  = + cot  = - y x

All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants: All Star TrigClass All functions are positive Sine is positive Tan is positiveCos is positive

reference angle The value of any trig function of an angle  is equal to the value of the corresponding trigonometric function of its reference angle, except possibly for the sign. The sign depends on the quadrant that  is in. So, now we know the signs of the trig functions, but what about their values?...

Reference Angles reference angle, The reference angle, α, is the angle between the terminal side and the nearest x -axis:

All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants: All Star TrigClass All functions are positive Sine is positive Tan is positiveCos is positive

Quadrantal Angles Quadrantal Angles ( terminal side lies along an axis )

Trig values of quadrantal angles:  0°0°90°180°270°360° 010– undefined –1 undefined 1 1 –1 undefined

Trigonometric Identities Reciprocal Identities Quotient Identities

Trigonometric Identities Pythagorean Identities  The fundamental Pythagorean identity:  Divide the first by sin 2 x :  Divide the first by cos 2 x :